Ë
    âQ(h…8 ã                   ó�  — d dl Z d dlmZ d dlmZmZ d dlZd dlZd dl	m
Z
 d dlmZ d dlmZmZmZ d dlmZ d dlmZ d d	lmZ d dlmZ d dlmc mZ d d
lmZmZ ddlm Z  ddl!m"Z#m$Z% ddl&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/ d dl0m1Z1 ddl2m3Z3m4Z4m5Z5 ddl6m7Z7m8Z8m9Z9m:Z:m;Z;m<Z<m=Z=m>Z> ddl?m@Z@ d dlAmBZB d dlCmDZD d dlEmFZF d„ ZGd„ ZH�d’d„ZI G d„ de+«      ZJ eJddd¬«      ZK G d„ de+«      ZL eLd ddd ¬!«      ZM G d"„ d#e+«      ZN eNdd$¬%«      ZO ej                   d&ej¢                  z  «      ZR ej¦                  eR«      ZTd'„ ZUd(„ ZVd)„ ZWd*„ ZXd+„ ZYd,„ ZZd-„ Z[d.„ Z\ G d/„ d0e+«      Z] e]d1¬2«      Z^ G d3„ d4e+«      Z_ e_dd5¬%«      Z` G d6„ d7e+«      Za eaej¢                   d8z  ej¢                  d8z  d9¬«      Zb G d:„ d;e+«      Zc ecddd<¬«      Zd G d=„ d>ee«      Zf G d?„ d@eD«      ZgdA„ ZhdB„ Zi G dC„ dDe+«      Zj ejdddE¬«      Zk G dF„ dGe+«      Zl elddH¬%«      Zm G dI„ dJe+«      Zn endddK¬«      Zo G dL„ dMe+«      Zp epddN¬%«      Zq G dO„ dPe+«      Zr erddQ¬%«      Zs G dR„ dSep«      Zt etddT¬%«      Zu G dU„ dVe+«      Zv evdW¬2«      Zw G dX„ dYe+«      Zx exddZ¬%«      Zy G d[„ d\e+«      Zz ezdd]¬%«      Z{ G d^„ d_e+«      Z| e|ej¢                   ej¢                  d`¬«      Z} G da„ dbe+«      Z~ e~dc¬2«      Z G dd„ dee+«      Z€ e€d df¬%«      Z� G dg„ dhe+«      Z‚ e‚di¬2«      Zƒ G dj„ dke+«      Z„ e„ddl¬%«      Z… G dm„ dne+«      Z† e†do¬2«      Z‡dp„ Zˆ G dq„ dre+«      Z‰ e‰dds¬%«      ZŠ G dt„ due+«      Z‹ e‹ddv¬%«      ZŒ G dw„ dxe+«      Z� e�ddy¬%«      ZŽ G dz„ d{e+«      Z� e�dd|¬%«      Z� G d}„ d~e+«      Z‘ e‘dd¬%«      Z’ G d€„ d�e+«      Z“ e“dd‚¬%«      Z” G dƒ„ d„e+«      Z• e•dd…¬%«      Z– G d†„ d‡e+«      Z— e—dˆ¬2«      Z˜d‰e˜_™         G dŠ„ d‹e+«      Zš ešddŒ¬�«      Z› G dŽ„ d�e+«      Zœ eœd�¬2«      Z� G d‘„ d’e+«      Zž eždd“¬%«      ZŸ G d”„ d•e+«      Z  e dd–¬%«      Z¡ G d—„ d˜e+«      Z¢ e¢d™¬2«      Z£dš„ Z¤ G d›„ dœe+«      Z¥ e¥dd�¬%«      Z¦ G dž„ dŸe¥«      Z§ e§dd ¬%«      Z¨ G d¡„ d¢e+«      Z© e©dd£¬%«      Zª G d¤„ d¥e+«      Z« e«dd¦¬%«      Z¬ G d§„ d¨e+«      Z­ e­d©¬2«      Z® G dª„ d«e+«      Z¯ e¯dd¬¬%«      Z°d­„ Z± G d®„ d¯e+«      Z² e²d°¬2«      Z³ G d±„ d²e+«      Z´ e´d³¬2«      Zµ G d´„ dµe+«      Z¶ e¶dd¶¬%«      Z· G d·„ d¸e+«      Z¸ e¸dd¹¬%«      Z¹ G dº„ d»e+«      Zº eºdd¼¬%«      Z» G d½„ d¾e+«      Z¼ e¼d¿¬2«      Z½ G dÀ„ dÁe+«      Z¾ e¾dddÂ¬«      Z¿ G dÃ„ dÄe+«      ZÀ eÀddÅ¬%«      ZÁ G dÆ„ dÇe+«      ZÂ eÂddÈ¬%«      ZÃ G dÉ„ dÊe+«      ZÄ eÄddË¬%«      ZÅ G dÌ„ dÍe+«      ZÆ eÆdÎ¬2«      ZÇ G dÏ„ dÐe+«      ZÈ eÈd dÑ¬%«      ZÉ G dÒ„ dÓe+«      ZÊ eÊdÔ¬2«      ZË G dÕ„ dÖe+«      ZÌ eÌddd×¬«      ZÍ G dØ„ dÙe+«      ZÎ eÎdÚ¬2«      ZÏ G dÛ„ dÜe+«      ZÐ eÐdÝ¬2«      ZÑ G dÞ„ dße+«      ZÒ eÒdà¬2«      ZÓ G dá„ dâe+«      ZÔ eÔdã¬2«      ZÕdä„ ZÖ G då„ dæe+«      Z× e×ddç¬%«      ZØ G dè„ dée+«      ZÙ eÙddê¬�«      ZÚ G dë„ dìe+«      ZÛ eÛdí¬2«      ZÜ G dî„ dïe+«      ZÝ eÝdð¬2«      ZÞ G dñ„ dòe+«      Zß eßddó¬%«      Zàdô„ Zá G dõ„ döe+«      Zâ eâdd÷¬%«      Zã G dø„ dùe+«      Zä eäddú¬%«      Zå G dû„ düe+«      Zæ eæddý¬%«      Zç G dþ„ dÿe+«      Zè eèd�d ¬%«      Zé G �d„ �de+«      Zê eê�d¬2«      Zë G �d„ �de+«      Zì eìd�d¬%«      Zí G �d„ �de+«      Zî eî�d	¬2«      Zï G �d
„ �de+«      Zð eðd�d¬%«      Zñ�d„ Zò G �d„ �de+«      Zó eód�d¬%«      Zô G �d„ �de+«      Zõ eõd�d¬%«      Zö G �d„ �de+«      Z÷ e÷�d¬2«      Zø G �d„ �de+«      Zù eù�d¬2«      Zú G �d„ �de+«      Zû eûd�d¬%«      Zü G �d„ �de+«      Zý eýd�d¬%«      Zþ G �d „ �d!e+«      Zÿ eÿ�d"¬2«      �Z  G �d#„ �d$e+«      �Z �edd�d%¬«      �Z G �d&„ �d'e+«      �Z �ed�d(¬%«      �Z G �d)„ �d*e+«      �Z �e�d+¬2«      �Z G �d,„ �d-e+«      �Z �e�d.d�d/¬«      �Z G �d0„ �d1e+«      �Z	 �e	d�d2¬%«      �Z
 G �d3„ �d4e+«      �Z �e�d5¬2«      �Z �e�d6¬2«      �Zd‰�e_™        d‰�e_™         G �d7„ �d8e+«      �Z �ed�d9¬%«      �Z G �d:„ �d;e+«      �Z �e�d<¬2«      �Z�d=�e_™         G �d>„ �d?e+«      �Z �ed�d@¬%«      �Z G �dA„ �dBe+«      �Z �e�d.d�dC¬«      �Z G �dD„ �dEe+«      �Z �e�dF¬2«      �Z G �dG„ �dHe+«      �Z �e�dI¬2«      �Z G �dJ„ �dKe+«      �Z�dL�Z G �dM„ �dN�e«      �Z �edd�dO¬«      �Z �edd�dP¬«      �Zg �dQ¢�Z G �dR„ �dS«      �Z �eD ]  �Z! �e"�e�e! �e �e!«      «       Œ  G �dT„ �dUe+«      �Z# �e#dd�dV¬«      �Z$ G �dW„ �dXe+«      �Z% �e%d�dY¬%«      �Z&�dZ�e&_™        �d[„ �Z'�d\„ �Z(�d]„ �Z) G �d^„ �d_e+«      �Z* �e*�d`d�¬a«      �Z+d‰�e+_™         G �db„ �dce+«      �Z, �e,d�dd¬%«      �Z-�de�e-_™         G �df„ �dge+«      �Z. �e.�dh¬2«      �Z/ G �di„ �djef«      �Z0 G �dk„ �dle+«      �Z1 �e1dd�dm¬«      �Z2 G �dn„ �doe+«      �Z3 �e3�dp¬2«      �Z4 �e3ej¢                   ej¢                  �dq¬«      �Z5 G �dr„ �dseÂ«      �Z6 �e6d�dt¬%«      �Z7 G �du„ �dve+«      �Z8 �e8dd&ej¢                  z  �dw¬«      �Z9 G �dx„ �dye+«      �Z: �e:�dz¬2«      �Z; G �d{„ �d|e+«      �Z< �e<d �d}¬%«      �Z= G �d~„ �de+«      �Z> �e>�d€�d��¬‚«      �Z?�dƒ„ �Z@ G �d„„ �d…e+«      �ZA �eA�d†�d‡dd�¬ˆ«      �ZB G �d‰„ �dŠe+«      �ZC G �d‹„ �dŒe+«      �ZD �eD�d�d e�jŠ                  �¬Ž«      �ZF G �d�„ �d�e+«      �ZG �eGd�d‘¬%«      �ZH �eI �eJ«       �j—                  «       �j™                  «       «      �ZM e(�eMe+«      \  �ZN�ZO�eN�eOz   �dŠgz   �ZPy(“  é    N)ÚIterable)ÚwrapsÚcached_property©Ú
Polynomial)ÚBSpline)Úextend_notes_in_docstringÚreplace_notes_in_docstringÚinherit_docstring_from)ÚLowLevelCallable)Úoptimize)Ú	integrate)Ú_lazyselectÚ
_lazywhereé   )Ú_stats)Útukeylambda_varianceÚtukeylambda_kurtosis)	Ú_vectorize_rvs_over_shapesÚget_distribution_namesÚ	_kurtosisÚ_isintegralÚrv_continuousÚ_skewÚ_get_fixed_fit_valueÚ_check_shapeÚ
_ShapeInfo)Ú	_log1mexp)ÚkolmognÚkolmognpÚkolmogni)Ú_XMINÚ_LOGXMINÚ_EULERÚ_ZETA3Ú_SQRT_PIÚ_SQRT_2_OVER_PIÚ_LOG_PIÚ_LOG_SQRT_2_OVER_PI)ÚCensoredData)Úroot_scalar)ÚFitErrorc                 ó¶   — | j                  dd«       | j                  dd«       | j                  dd«       | j                  dd«       | rt        d| › d�«      ‚y)a†  
    Remove the optimizer-related keyword arguments 'loc', 'scale' and
    'optimizer' from `kwds`.  Then check that `kwds` is empty, and
    raise `TypeError("Unknown arguments: %s." % kwds)` if it is not.

    This function is used in the fit method of distributions that override
    the default method and do not use the default optimization code.

    `kwds` is modified in-place.
    ÚlocNÚscaleÚ	optimizerÚmethodzUnknown arguments: ú.)ÚpopÚ	TypeError)Úkwdss    ú\/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/scipy/stats/_continuous_distns.pyÚ_remove_optimizer_parametersr7   '   sY   € ð 	‡H�HˆU�DÔØ‡H�HˆW�dÔØ‡H�Hˆ[˜$ÔØ‡H�HˆX�tÔÙÜÐ-¨d¨V°1Ð5Ó6Ð6ð ó    c                 ó.   ‡ — t        ‰ «      ˆ fd„«       }|S )Nc                 ó  •— |j                  dd«      j                  «       }t        |t        «      }|dk(  s|r/|j	                  «       dkD  rt        t        | «      | �  |g|¢­i |¤ŽS |r|j                  } ‰| |g|¢­i |¤ŽS )Nr1   ÚmleÚmmr   )	ÚgetÚlowerÚ
isinstancer*   Únum_censoredÚsuperÚtypeÚfitÚ_uncensored)ÚselfÚdataÚargsr5   r1   ÚcensoredÚfuns         €r6   Úwrapperz _call_super_mom.<locals>.wrapper>   sˆ   ø€ à—‘˜( EÓ*×0Ñ0Ó2ˆÜ˜d¤LÓ1ˆØ�TŠ>™h¨4×+<Ñ+<Ó+>ÀÒ+BÜœ˜d› TÑ.¨tÐC°dÒC¸dÑCÐCáð ×'Ñ'�Ù�t˜TÐ1 DÒ1¨DÑ1Ð1r8   )r   )rI   rJ   s   ` r6   Ú_call_super_momrK   :   s"   ø€ ô ˆ3ƒZó2ó ð2ð €Nr8   c                 ó¬   ‡ — |xs |dz
  }||z
  }ˆ fd„} |||«      s6|dz  }||z
  }d}t        j                  |«      rt        |«      ‚ |||«      sŒ6|S )Nr   c                 ór   •— t        j                   ‰| «      «      t        j                   ‰|«      «      k7  S ©N©ÚnpÚsign)ÚlbrackÚrbrackrI   s     €r6   Úinterval_contains_rootz1_get_left_bracket.<locals>.interval_contains_rootV   s(   ø€ ä�w‰w‘s˜6“{Ó#¤r§w¡w©s°6«{Ó';Ñ;Ð;r8   é   zVThe solver could not find a bracket containing a root to an MLE first order condition.)rP   ÚisinfÚFitSolverError)rI   rS   rR   ÚdiffrT   Úmsgs   `     r6   Ú_get_left_bracketrZ   O   sn   ø€ àÒ!�v ‘z€FØ�F‰?€Dô<ñ % V¨VÔ4Ø�‰	ˆØ˜$‘ˆð7ˆä�8‰8�FÔÜ  Ó%Ð%ñ % V¨VÕ4ð €Mr8   c                   ó:   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
y	)
Ú	ksone_genaá  Kolmogorov-Smirnov one-sided test statistic distribution.

    This is the distribution of the one-sided Kolmogorov-Smirnov (KS)
    statistics :math:`D_n^+` and :math:`D_n^-`
    for a finite sample size ``n >= 1`` (the shape parameter).

    %(before_notes)s

    See Also
    --------
    kstwobign, kstwo, kstest

    Notes
    -----
    :math:`D_n^+` and :math:`D_n^-` are given by

    .. math::

        D_n^+ &= \text{sup}_x (F_n(x) - F(x)),\\
        D_n^- &= \text{sup}_x (F(x) - F_n(x)),\\

    where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
    `ksone` describes the distribution under the null hypothesis of the KS test
    that the empirical CDF corresponds to :math:`n` i.i.d. random variates
    with CDF :math:`F`.

    %(after_notes)s

    References
    ----------
    .. [1] Birnbaum, Z. W. and Tingey, F.H. "One-sided confidence contours
       for probability distribution functions", The Annals of Mathematical
       Statistics, 22(4), pp 592-596 (1951).

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import ksone
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)

    Display the probability density function (``pdf``):

    >>> n = 1e+03
    >>> x = np.linspace(ksone.ppf(0.01, n),
    ...                 ksone.ppf(0.99, n), 100)
    >>> ax.plot(x, ksone.pdf(x, n),
    ...         'r-', lw=5, alpha=0.6, label='ksone pdf')

    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.

    Freeze the distribution and display the frozen ``pdf``:

    >>> rv = ksone(n)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    Check accuracy of ``cdf`` and ``ppf``:

    >>> vals = ksone.ppf([0.001, 0.5, 0.999], n)
    >>> np.allclose([0.001, 0.5, 0.999], ksone.cdf(vals, n))
    True

    c                 ó>   — |dk\  |t        j                  |«      k(  z  S ©Nr   ©rP   Úround©rE   Úns     r6   Ú	_argcheckzksone_gen._argcheckª   ó   € Ø�Q‘˜1¤§¡¨£Ñ+Ñ,Ð,r8   c                 ó@   — t        dddt        j                  fd«      gS ©Nrb   Tr   ©TF©r   rP   Úinf©rE   s    r6   Ú_shape_infozksone_gen._shape_info­   ó   € Ü˜3  q¬"¯&©& k°=ÓAÐBÐBr8   c                 ó0   — t        j                  ||«       S rN   )ÚscuÚ	_smirnovp©rE   Úxrb   s      r6   Ú_pdfzksone_gen._pdf°   s   € Ü—‘˜a Ó#Ð#Ð#r8   c                 ó.   — t        j                  ||«      S rN   )rn   Ú	_smirnovcrp   s      r6   Ú_cdfzksone_gen._cdf³   s   € Ü�}‰}˜Q Ó"Ð"r8   c                 ó.   — t        j                  ||«      S rN   )ÚscÚsmirnovrp   s      r6   Ú_sfzksone_gen._sf¶   s   € Ü�z‰z˜!˜QÓÐr8   c                 ó.   — t        j                  ||«      S rN   )rn   Ú
_smirnovci©rE   Úqrb   s      r6   Ú_ppfzksone_gen._ppf¹   s   € Ü�~‰~˜a Ó#Ð#r8   c                 ó.   — t        j                  ||«      S rN   )rw   Úsmirnovir|   s      r6   Ú_isfzksone_gen._isf¼   ó   € Ü�{‰{˜1˜aÓ Ð r8   N)Ú__name__Ú
__module__Ú__qualname__Ú__doc__rc   rk   rr   ru   ry   r~   r�   © r8   r6   r\   r\   f   s-   „ ñBòF-òCò$ò#ò ò$ó!r8   r\   ç        ç      ð?Úksone)ÚaÚbÚnamec                   ó@   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zy
)Ú	kstwo_genad  Kolmogorov-Smirnov two-sided test statistic distribution.

    This is the distribution of the two-sided Kolmogorov-Smirnov (KS)
    statistic :math:`D_n` for a finite sample size ``n >= 1``
    (the shape parameter).

    %(before_notes)s

    See Also
    --------
    kstwobign, ksone, kstest

    Notes
    -----
    :math:`D_n` is given by

    .. math::

        D_n = \text{sup}_x |F_n(x) - F(x)|

    where :math:`F` is a (continuous) CDF and :math:`F_n` is an empirical CDF.
    `kstwo` describes the distribution under the null hypothesis of the KS test
    that the empirical CDF corresponds to :math:`n` i.i.d. random variates
    with CDF :math:`F`.

    %(after_notes)s

    References
    ----------
    .. [1] Simard, R., L'Ecuyer, P. "Computing the Two-Sided
       Kolmogorov-Smirnov Distribution",  Journal of Statistical Software,
       Vol 39, 11, 1-18 (2011).

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import kstwo
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)

    Display the probability density function (``pdf``):

    >>> n = 10
    >>> x = np.linspace(kstwo.ppf(0.01, n),
    ...                 kstwo.ppf(0.99, n), 100)
    >>> ax.plot(x, kstwo.pdf(x, n),
    ...         'r-', lw=5, alpha=0.6, label='kstwo pdf')

    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.

    Freeze the distribution and display the frozen ``pdf``:

    >>> rv = kstwo(n)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    Check accuracy of ``cdf`` and ``ppf``:

    >>> vals = kstwo.ppf([0.001, 0.5, 0.999], n)
    >>> np.allclose([0.001, 0.5, 0.999], kstwo.cdf(vals, n))
    True

    c                 ó>   — |dk\  |t        j                  |«      k(  z  S r^   r_   ra   s     r6   rc   zkstwo_gen._argcheck  rd   r8   c                 ó@   — t        dddt        j                  fd«      gS rf   rh   rj   s    r6   rk   zkstwo_gen._shape_info	  rl   r8   c                 ób   — dt        |t        «      s|z  dfS t        j                  |«      z  dfS ©Nç      à?r‰   )r?   r   rP   Ú
asanyarrayra   s     r6   Ú_get_supportzkstwo_gen._get_support  s;   € Øœj¨¬HÔ5�QÑLØðð 	¼2¿=¹=ÈÓ;KÑLØðð 	r8   c                 ó   — t        ||«      S rN   )r    rp   s      r6   rr   zkstwo_gen._pdf  s   € Ü˜˜1‹~Ðr8   c                 ó   — t        ||«      S rN   ©r   rp   s      r6   ru   zkstwo_gen._cdf  s   € Ü�q˜!‹}Ðr8   c                 ó   — t        ||d¬«      S ©NF©Úcdfr™   rp   s      r6   ry   zkstwo_gen._sf  s   € Ü�q˜! Ô'Ð'r8   c                 ó   — t        ||d¬«      S )NTrœ   ©r!   r|   s      r6   r~   zkstwo_gen._ppf  s   € Ü˜˜1 $Ô'Ð'r8   c                 ó   — t        ||d¬«      S r›   rŸ   r|   s      r6   r�   zkstwo_gen._isf  s   € Ü˜˜1 %Ô(Ð(r8   N)rƒ   r„   r…   r†   rc   rk   r–   rr   ru   ry   r~   r�   r‡   r8   r6   r�   r�   Ã   s2   „ ñAòD-òCòòòò(ò(ó)r8   r�   Úkstwo)Úmomtyper‹   rŒ   r�   c                   ó4   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	y)	Úkstwobign_gena  Limiting distribution of scaled Kolmogorov-Smirnov two-sided test statistic.

    This is the asymptotic distribution of the two-sided Kolmogorov-Smirnov
    statistic :math:`\sqrt{n} D_n` that measures the maximum absolute
    distance of the theoretical (continuous) CDF from the empirical CDF.
    (see `kstest`).

    %(before_notes)s

    See Also
    --------
    ksone, kstwo, kstest

    Notes
    -----
    :math:`\sqrt{n} D_n` is given by

    .. math::

        D_n = \text{sup}_x |F_n(x) - F(x)|

    where :math:`F` is a continuous CDF and :math:`F_n` is an empirical CDF.
    `kstwobign`  describes the asymptotic distribution (i.e. the limit of
    :math:`\sqrt{n} D_n`) under the null hypothesis of the KS test that the
    empirical CDF corresponds to i.i.d. random variates with CDF :math:`F`.

    %(after_notes)s

    References
    ----------
    .. [1] Feller, W. "On the Kolmogorov-Smirnov Limit Theorems for Empirical
       Distributions",  Ann. Math. Statist. Vol 19, 177-189 (1948).

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zkstwobign_gen._shape_infoI  ó   € Øˆ	r8   c                 ó.   — t        j                  |«       S rN   )rn   Ú_kolmogp©rE   rq   s     r6   rr   zkstwobign_gen._pdfL  s   € Ü—‘˜Q“ÐÐr8   c                 ó,   — t        j                  |«      S rN   )rn   Ú_kolmogcr©   s     r6   ru   zkstwobign_gen._cdfO  s   € Ü�|‰|˜A‹Ðr8   c                 ó,   — t        j                  |«      S rN   )rw   Ú
kolmogorovr©   s     r6   ry   zkstwobign_gen._sfR  s   € Ü�}‰}˜QÓÐr8   c                 ó,   — t        j                  |«      S rN   )rn   Ú	_kolmogci©rE   r}   s     r6   r~   zkstwobign_gen._ppfU  s   € Ü�}‰}˜QÓÐr8   c                 ó,   — t        j                  |«      S rN   )rw   Úkolmogir°   s     r6   r�   zkstwobign_gen._isfX  s   € Ü�z‰z˜!‹}Ðr8   N)
rƒ   r„   r…   r†   rk   rr   ru   ry   r~   r�   r‡   r8   r6   r¤   r¤   $  s&   „ ñ#òHò òò ò ór8   r¤   Ú	kstwobign)r‹   r�   rU   c                 óH   — t        j                  | dz   dz  «      t        z  S ©NrU   ç       @)rP   ÚexpÚ_norm_pdf_C©rq   s    r6   Ú	_norm_pdfrº   h  s    € Ü�6‰6�1�a‘4�%˜‘)Óœ{Ñ*Ð*r8   c                 ó"   — | dz   dz  t         z
  S rµ   )Ú_norm_pdf_logCr¹   s    r6   Ú_norm_logpdfr½   l  s   € Øˆq‰Dˆ5�3‰;œÑ'Ð'r8   c                 ó,   — t        j                  | «      S rN   )rw   Úndtrr¹   s    r6   Ú	_norm_cdfrÀ   p  s   € Ü�7‰7�1‹:Ðr8   c                 ó,   — t        j                  | «      S rN   )rw   Úlog_ndtrr¹   s    r6   Ú_norm_logcdfrÃ   t  s   € Ü�;‰;�q‹>Ðr8   c                 ó,   — t        j                  | «      S rN   )rw   Úndtri©r}   s    r6   Ú	_norm_ppfrÇ   x  s   € Ü�8‰8�A‹;Ðr8   c                 ó   — t        |  «      S rN   ©rÀ   r¹   s    r6   Ú_norm_sfrÊ   |  s   € Ü�a�R‹=Ðr8   c                 ó   — t        |  «      S rN   ©rÃ   r¹   s    r6   Ú_norm_logsfrÍ   €  s   € Ü˜˜ÓÐr8   c                 ó   — t        | «       S rN   ©rÇ   rÆ   s    r6   Ú	_norm_isfrÐ   „  s   € Ü�a‹Lˆ=Ðr8   c                   óŠ   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Zd„ Zd„ Zd„ Ze eed¬«      d„ «       «       Zd„ Zy)Únorm_gena�  A normal continuous random variable.

    The location (``loc``) keyword specifies the mean.
    The scale (``scale``) keyword specifies the standard deviation.

    %(before_notes)s

    Notes
    -----
    The probability density function for `norm` is:

    .. math::

        f(x) = \frac{\exp(-x^2/2)}{\sqrt{2\pi}}

    for a real number :math:`x`.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   znorm_gen._shape_infoŸ  r¦   r8   Nc                 ó$   — |j                  |«      S rN   )Ústandard_normal©rE   ÚsizeÚrandom_states      r6   Ú_rvsznorm_gen._rvs¢  s   € Ø×+Ñ+¨DÓ1Ð1r8   c                 ó   — t        |«      S rN   ©rº   r©   s     r6   rr   znorm_gen._pdf¥  s   € ä˜‹|Ðr8   c                 ó   — t        |«      S rN   ©r½   r©   s     r6   Ú_logpdfznorm_gen._logpdf©  ó   € Ü˜A‹Ðr8   c                 ó   — t        |«      S rN   rÉ   r©   s     r6   ru   znorm_gen._cdf¬  ó   € Ü˜‹|Ðr8   c                 ó   — t        |«      S rN   rÌ   r©   s     r6   Ú_logcdfznorm_gen._logcdf¯  rß   r8   c                 ó   — t        |«      S rN   ©rÊ   r©   s     r6   ry   znorm_gen._sf²  s   € Ü˜‹{Ðr8   c                 ó   — t        |«      S rN   )rÍ   r©   s     r6   Ú_logsfznorm_gen._logsfµ  s   € Ü˜1‹~Ðr8   c                 ó   — t        |«      S rN   rÏ   r°   s     r6   r~   znorm_gen._ppf¸  rá   r8   c                 ó   — t        |«      S rN   ©rÐ   r°   s     r6   r�   znorm_gen._isf»  rá   r8   c                  ó   — y)N)rˆ   r‰   rˆ   rˆ   r‡   rj   s    r6   r   znorm_gen._stats¾  ó   € Ø!r8   c                 óZ   — dt        j                  dt         j                  z  «      dz   z  S ©Nr”   rU   r   ©rP   ÚlogÚpirj   s    r6   Ú_entropyznorm_gen._entropyÁ  s"   € Ø”B—F‘F˜1œRŸU™U™7“O AÑ%Ñ&Ð&r8   a}          For the normal distribution, method of moments and maximum likelihood
        estimation give identical fits, and explicit formulas for the estimates
        are available.
        This function uses these explicit formulas for the maximum likelihood
        estimation of the normal distribution parameters, so the
        `optimizer` and `method` arguments are ignored.

©Únotesc                 ó˜  — |j                  dd «      }|j                  dd «      }t        |«       |�|�t        d«      ‚t        j                  |«      }t        j
                  |«      j                  «       st        d«      ‚|€|j                  «       }n|}|€-t        j                  ||z
  dz  j                  «       «      }||fS |}||fS )NÚflocÚfscaleú3All parameters fixed. There is nothing to optimize.ú$The data contains non-finite values.rU   )	r3   r7   Ú
ValueErrorrP   ÚasarrayÚisfiniteÚallÚmeanÚsqrt)rE   rF   r5   rö   r÷   r.   r/   s          r6   rC   znorm_gen.fitÄ  sÒ   € ð �x‰x˜ Ó%ˆØ—‘˜( DÓ)ˆä$ TÔ*àÐ Ð 2ô ð )ó *ð *ô �z‰z˜$Óˆä�{‰{˜4Ó ×$Ñ$Ô&ÜÐCÓDÐDàˆ<Ø—)‘)“+‰CàˆCàˆ>Ü—G‘G˜d S™j¨1™_×2Ñ2Ó4Ó5ˆEð �EˆzÐð ˆEà�EˆzÐr8   c                 ób   — |dk(  ry|dz  dk(  r!t        j                  t        |«      dz
  «      S y)zŽ
        @returns Moments of standard normal distribution for integer n >= 0

        See eq. 16 of https://arxiv.org/abs/1209.4340v2
        r   r‰   rU   r   rˆ   )rw   Ú
factorial2Úintra   s     r6   Ú_munpznorm_gen._munpê  s3   € ð �Š6ØØˆq‰5�AŠ:Ü—=‘=¤ Q£¨!¡Ó,Ð,àr8   ©NN)rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   rã   ry   rç   r~   r�   r   rò   rK   r
   r   rC   r  r‡   r8   r6   rÒ   rÒ   ˆ  sr   „ ñò,ó2òòòòòòòòò"ò'ð Ù ð 6?ô @ñó@ó ðó<r8   rÒ   Únorm)r�   c                   óL   — e Zd ZdZej
                  Zd„ Zd„ Zd„ Z	d„ Z
d„ Zd„ Zy)	Ú	alpha_gena&  An alpha continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `alpha` ([1]_, [2]_) is:

    .. math::

        f(x, a) = \frac{1}{x^2 \Phi(a) \sqrt{2\pi}} *
                  \exp(-\frac{1}{2} (a-1/x)^2)

    where :math:`\Phi` is the normal CDF, :math:`x > 0`, and :math:`a > 0`.

    `alpha` takes ``a`` as a shape parameter.

    %(after_notes)s

    References
    ----------
    .. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate
           Distributions, Volume 1", Second Edition, John Wiley and Sons,
           p. 173 (1994).
    .. [2] Anthony A. Salvia, "Reliability applications of the Alpha
           Distribution", IEEE Transactions on Reliability, Vol. R-34,
           No. 3, pp. 251-252 (1985).

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS ©Nr‹   Fr   ©FFrh   rj   s    r6   rk   zalpha_gen._shape_info  ó   € Ü˜3 ¨¬2¯6©6 {°NÓCÐDÐDr8   c                 óN   — d|dz  z  t        |«      z  t        |d|z  z
  «      z  S ©Nr‰   rU   )rÀ   rº   ©rE   rq   r‹   s      r6   rr   zalpha_gen._pdf   s+   € à�A�q‘D‰zœ) A›,Ñ&¤y°°3°q±5±Ó'9Ñ9Ð9r8   c                 ó”   — dt        j                  |«      z  t        |d|z  z
  «      z   t        j                  t        |«      «      z
  S )Néþÿÿÿr‰   )rP   rð   r½   rÀ   r  s      r6   rÞ   zalpha_gen._logpdf$  s8   € Ø”"—&‘&˜“)‰|œl¨1¨S°©U©7Ó3Ñ3´b·f±f¼YÀq»\Ó6JÑJÐJr8   c                 ó<   — t        |d|z  z
  «      t        |«      z  S ©Nr‰   rÉ   r  s      r6   ru   zalpha_gen._cdf'  s   € Ü˜˜3˜q™5™Ó!¤I¨a£LÑ0Ð0r8   c           
      ób   — dt        j                  |t        |t        |«      z  «      z
  «      z  S r  )rP   rû   rÇ   rÀ   ©rE   r}   r‹   s      r6   r~   zalpha_gen._ppf*  s(   € Ø”2—:‘:˜a¤)¨A¬i¸«l©NÓ";Ñ;Ó<Ñ<Ð<r8   c                 óT   — t         j                  gdz  t         j                  gdz  z   S ©NrU   ©rP   ri   Únan©rE   r‹   s     r6   r   zalpha_gen._stats-  s!   € Ü—‘ˆx˜‰zœRŸV™V˜H Q™JÑ&Ð&r8   N)rƒ   r„   r…   r†   r   Ú_open_support_maskÚ_support_maskrk   rr   rÞ   ru   r~   r   r‡   r8   r6   r  r  û  s4   „ ñð> "×4Ñ4€MòEò:òKò1ò=ó'r8   r  Úalphac                   ó:   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
y	)
Ú
anglit_gena  An anglit continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `anglit` is:

    .. math::

        f(x) = \sin(2x + \pi/2) = \cos(2x)

    for :math:`-\pi/4 \le x \le \pi/4`.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zanglit_gen._shape_infoH  r¦   r8   c                 ó2   — t        j                  d|z  «      S r  )rP   Úcosr©   s     r6   rr   zanglit_gen._pdfK  s   € ä�v‰v�a˜‘c‹{Ðr8   c                 óZ   — t        j                  |t         j                  dz  z   «      dz  S ©Né   r¶   ©rP   Úsinrñ   r©   s     r6   ru   zanglit_gen._cdfO  s"   € Ü�v‰v�aœŸ™˜a™‘iÓ  #Ñ%Ð%r8   c                 óZ   — t        j                  |t         j                  dz  z   «      dz  S r#  )rP   r!  rñ   r©   s     r6   ry   zanglit_gen._sfR  s"   € Ü�v‰v�aœ"Ÿ%™% !™)‘mÓ$¨Ñ+Ð+r8   c                 óz   — t        j                  t        j                  |«      «      t         j                  dz  z
  S ©Nr$  )rP   Úarcsinrÿ   rñ   r°   s     r6   r~   zanglit_gen._ppfU  s&   € Ü�y‰yœŸ™ ›Ó$¤R§U¡U¨1¡WÑ,Ð,r8   c                 óÖ   — dt         j                  t         j                  z  dz  dz
  ddt         j                  dz  dz
  z  t         j                  t         j                  z  dz
  dz  z  fS )	Nrˆ   é   r”   r  r$  é`   é   rU   ©rP   rñ   rj   s    r6   r   zanglit_gen._statsX  sR   € Ø”B—E‘Eœ"Ÿ%™%‘K ‘N 3Ñ&¨¨R´·±¸±¸B±Ñ-?ÄÇÁÄrÇuÁuÁÈQÁÐQRÑ@RÑ-RÐRÐRr8   c                 ó2   — dt        j                  d«      z
  S ©Nr   rU   ©rP   rð   rj   s    r6   rò   zanglit_gen._entropy[  ó   € Ø”—‘˜“‰{Ðr8   N)rƒ   r„   r…   r†   rk   rr   ru   ry   r~   r   rò   r‡   r8   r6   r  r  4  s+   „ ñò&òò&ò,ò-òSór8   r  r$  Úanglitc                   ó4   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	y)	Úarcsine_gena  An arcsine continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `arcsine` is:

    .. math::

        f(x) = \frac{1}{\pi \sqrt{x (1-x)}}

    for :math:`0 < x < 1`.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zarcsine_gen._shape_infov  r¦   r8   c                 ó¸   — t        j                  d¬«      5  dt         j                  z  t        j                  |d|z
  z  «      z  cd d d «       S # 1 sw Y   y xY w)NÚignore©Údivider‰   r   )rP   Úerrstaterñ   rÿ   r©   s     r6   rr   zarcsine_gen._pdfy  sD   € ä�[‰[ Ô)ñ 	.Ø”r—u‘u‘9œRŸW™W Q¨¨!©¡WÓ-Ñ-÷	.÷ 	.ò 	.ús   —/AÁAc                 óz   — dt         j                  z  t        j                  t        j                  |«      «      z  S ©Nr¶   )rP   rñ   r*  rÿ   r©   s     r6   ru   zarcsine_gen._cdf~  s&   € Ø”2—5‘5‰yœŸ™¤2§7¡7¨1£:Ó.Ñ.Ð.r8   c                 óZ   — t        j                  t         j                  dz  |z  «      dz  S r>  r%  r°   s     r6   r~   zarcsine_gen._ppf�  s"   € Ü�v‰v”b—e‘e˜C‘i ‘kÓ" CÑ'Ð'r8   c                 ó   — d}d}d}d}||||fS )Nr”   g      À?r   ç      ø¿r‡   ©rE   ÚmuÚmu2Úg1Úg2s        r6   r   zarcsine_gen._stats„  s$   € ØˆØˆØˆØˆØ�3˜˜BˆÐr8   c                  ó   — y)Ng‘Á”°•ëÎ¿r‡   rj   s    r6   rò   zarcsine_gen._entropy‹  s   € Ø&r8   N©
rƒ   r„   r…   r†   rk   rr   ru   r~   r   rò   r‡   r8   r6   r6  r6  b  s%   „ ñò&ò.ò
/ò(òó'r8   r6  Úarcsinec                   ó   — e Zd ZdZd„ Zy)ÚFitDataErrorz=Raised when input data is inconsistent with fixed parameters.c                 ó(   — d|›d|›d|›d�f| _         y )Nz>Invalid values in `data`.  Maximum likelihood estimation with z requires that z < (x - loc)/scale  < z for each x in `data`.©rG   )rE   Údistrr>   Úuppers       r6   Ú__init__zFitDataError.__init__—  s/   € ðØ$˜i °u°ið @"Ø"' Ð*@ðBð
ˆ�	r8   N©rƒ   r„   r…   r†   rP  r‡   r8   r6   rK  rK  ’  s
   „ ÙGó
r8   rK  c                   ó   — e Zd ZdZd„ Zy)rW   zN
    Raised when a solver fails to converge while fitting a distribution.
    c                 óB   — d}||j                  dd«      z  }|f| _        y )Nz1Solver for the MLE equations failed to converge: ú
Ú )ÚreplacerG   )rE   ÚmesgÚemsgs      r6   rP  zFitSolverError.__init__¥  s%   € ØBˆØ�—‘˜T 2Ó&Ñ&ˆØ�Gˆ�	r8   NrQ  r‡   r8   r6   rW   rW   Ÿ  s   „ ñó
r8   rW   c                 ót   — t        j                  | |z   «      }||| t        j                  | «      z   z  z
  }|S rN   ©rw   Úpsi)r‹   rŒ   rb   Ús1ÚpsiabÚfuncs         r6   Ú_beta_mle_ar_  «  s8   € ô �F‰F�1�q‘5‹M€EØ��e�VœbŸf™f Q›iÑ'Ñ(Ñ(€DØ€Kr8   c                 ó¼   — | \  }}t        j                  ||z   «      }||| t        j                  |«      z   z  z
  ||| t        j                  |«      z   z  z
  g}|S rN   rZ  )Úthetarb   r\  Ús2r‹   rŒ   r]  r^  s           r6   Ú_beta_mle_abrc  ´  sb   € ð �D€A€qÜ�F‰F�1�q‘5‹M€EØ��u�fœrŸv™v a›yÑ(Ñ)Ñ)Ø��u�fœrŸv™v a›yÑ(Ñ)Ñ)ð+€Dà€Kr8   c                   óŽ   ‡ — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zˆ fd„Ze eed¬«      ˆ fd„«       «       Zd„ Zˆ xZS )Úbeta_genaÿ  A beta continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `beta` is:

    .. math::

        f(x, a, b) = \frac{\Gamma(a+b) x^{a-1} (1-x)^{b-1}}
                          {\Gamma(a) \Gamma(b)}

    for :math:`0 <= x <= 1`, :math:`a > 0`, :math:`b > 0`, where
    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `beta` takes :math:`a` and :math:`b` as shape parameters.

    This distribution uses routines from the Boost Math C++ library for
    the computation of the ``pdf``, ``cdf``, ``ppf``, ``sf`` and ``isf``
    methods. [1]_

    %(after_notes)s

    References
    ----------
    .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

    %(example)s

    c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS ©Nr‹   Fr   r
  rŒ   rh   ©rE   ÚiaÚibs      r6   rk   zbeta_gen._shape_infoâ  ó;   € Ü˜˜U Q¬¯© K°Ó@ˆÜ˜˜U Q¬¯© K°Ó@ˆØ�Bˆxˆr8   c                 ó(   — |j                  |||«      S rN   ©Úbeta)rE   r‹   rŒ   r×   rØ   s        r6   rÙ   zbeta_gen._rvsç  s   € Ø× Ñ   A tÓ,Ð,r8   c                 óˆ   — t        j                  d¬«      5  t        j                  |||«      cd d d «       S # 1 sw Y   y xY w©Nr9  ©Úover)rP   r<  rn   Ú	_beta_pdf©rE   rq   r‹   rŒ   s       r6   rr   zbeta_gen._pdfê  s5   € ô �[‰[˜hÔ'ñ 	*Ü—=‘=  A qÓ)÷	*÷ 	*ò 	*úó	   —8¸Ac                 ó    — t        j                  |dz
  | «      t        j                  |dz
  |«      z   }|t        j                  ||«      z  }|S r  )rw   Úxlog1pyÚxlogyÚbetaln)rE   rq   r‹   rŒ   ÚlPxs        r6   rÞ   zbeta_gen._logpdfñ  sE   € Ü�j‰j˜˜S™ 1 "Ó%¬¯©°°S±¸!Ó(<Ñ<ˆØŒr�y‰y˜˜A‹ÑˆØˆ
r8   c                 ó0   — t        j                  |||«      S rN   )rw   Úbetaincrt  s       r6   ru   zbeta_gen._cdfö  s   € Ü�z‰z˜!˜Q Ó"Ð"r8   c                 ó0   — t        j                  |||«      S rN   )rw   Úbetainccrt  s       r6   ry   zbeta_gen._sfù  s   € Ü�{‰{˜1˜a Ó#Ð#r8   c                 ó0   — t        j                  |||«      S rN   )rw   Úbetainccinvrt  s       r6   r�   zbeta_gen._isfü  s   € Ü�~‰~˜a  AÓ&Ð&r8   c                 ó0   — t        j                  |||«      S rN   )rn   Ú	_beta_ppf©rE   r}   r‹   rŒ   s       r6   r~   zbeta_gen._ppfÿ  s   € Ü�}‰}˜Q  1Ó%Ð%r8   c                 ó*  — ||z   }||z  }||z  |dz  |dz   z  z  }d||z
  z  t        j                  |dz   «      z  |dz   t        j                  ||z  «      z  z  }d||z
  dz  |dz   z  ||z  |dz   z  z
  z  }||z  |dz   z  |dz   z  }||z  }	||||	fS )NrU   r   é   é   ©rP   rÿ   )
rE   r‹   rŒ   Úa_plus_bÚ
_beta_meanÚ_beta_varianceÚ_beta_skewnessÚ_beta_kurtosis_excess_nÚ_beta_kurtosis_excess_dÚ_beta_kurtosis_excesss
             r6   r   zbeta_gen._stats  sÜ   € Ø�q‘5ˆØ�x‘Zˆ
Ø˜1™ ¨!¡¨x¸!©|Ñ <Ñ=ˆØ  A¡™;¬¯©°¸A±Ó)>Ñ>Ø$ q™L¬B¯G©G°A¸±E«NÑ:ñ<ˆà"#¨¨A©°¡z°XÀ±\Ñ'BØ'(¨1¡u°¸1±Ñ'=ñ(>ñ #?Ðà"# a¡%¨8°a©<Ñ"8¸HÀq¹LÑ"IÐØ 7Ð:QÑ QÐàØØØ!ð	#ð 	#r8   c                 óØ   •‡‡— t        |t        «      r|j                  «       }t        |«      Št	        |«      Šˆˆfd„}t        j                  |d«      \  }}t        ‰| �!  |||f¬«      S )Nc                 óJ  •— | \  }}d||z
  z  t        j                  ||z   dz   «      z  ||z   dz   z  t        j                  ||z  «      z  }|dz  |dz  d|z  dz
  z  z
  |dz  |dz   z  z   d|z  |z  |dz   z  z
  }|||z  ||z   dz   z  ||z   dz   z  z  }|dz  }|‰z
  |‰z
  gS )NrU   r   r†  r…  r‡  )rq   r‹   rŒ   ÚskÚkurE  rF  s        €€r6   r^  z beta_gen._fitstart.<locals>.func  sÓ   ø€ Ø‰DˆAˆqØ�A�a‘C‘œŸ™  Q¡¨¡Ó+Ñ+¨q°1©u°q©yÑ9¼B¿G¹GÀAÀaÁC»LÑHˆBØ�A‘˜˜1™˜a ™c !™e™Ñ$ q¨!¡t¨Q¨q©S¡zÑ1°A°a±C¸±E¸1¸Q¹3±KÑ?ˆBØ�!�A‘#�q˜‘s˜1‘u‘+˜q ™s 1™uÑ%Ñ%ˆBØ�!‰GˆBØ�r‘E˜2˜b™5�>Ð!r8   )r‰   r‰   rM  )	r?   r*   Ú	_uncensorr   r   r   ÚfsolverA   Ú	_fitstart)rE   rF   r^  r‹   rŒ   rE  rF  Ú	__class__s        @@€r6   r•  zbeta_gen._fitstart  s_   ú€ Ü�dœLÔ)Ø—>‘>Ó#ˆDä�4‹[ˆÜ�t‹_ˆõ	"ô �‰˜t ZÓ0‰ˆˆ1Ü‰wÑ  ¨Q°¨FÐ Ó3Ð3r8   zÓ        In the special case where `method="MLE"` and
        both `floc` and `fscale` are given, a
        `ValueError` is raised if any value `x` in `data` does not satisfy
        `floc < x < floc + fscale`.

ró   c           	      óö  •— |j                  dd «      }|j                  dd «      }|�|€t        ‰| �  |g|¢­i |¤ŽS |j                  dd «       |j                  dd «       t	        |g d¢«      }t	        |g d¢«      }t        |«       |�|�t        d«      ‚t        j                  |«      j                  «       st        d«      ‚t        j                  |«      |z
  |z  }t        j                  |dk  «      st        j                  |dk\  «      rt        d	|||z   ¬
«      ‚|j                  «       }|€|�ˆ|�|}	d|z
  }d|z
  }n|}	|	|z  d|z
  z  }
t        j                  t         |
|	t#        |«      t        j$                  |«      j'                  «       fd¬«      \  }}}}|dk7  rt)        |¬«      ‚|d   }
|�½|	|
}	}
n¸t        j$                  |«      j'                  «       }t+        j,                  | «      j'                  «       }|d|z
  z  |j/                  d¬«      z  dz
  }||z  }
d|z
  |z  }	t        j                  t0        |
|	gt#        |«      ||fd¬«      \  }}}}|dk7  rt)        |¬«      ‚|\  }
}	|
|	||fS )Nrö   r÷   ©Úf0ÚfaÚfix_a)Úf1ÚfbÚfix_brø   rù   r   r   rn  ©r>   rO  T)rG   Úfull_output)rW  )Úddof)r=   rA   rC   r3   r   r7   rú   rP   rü   rý   ÚravelÚanyrK  rþ   r   r”  r_  Úlenrð   ÚsumrW   rw   Úlog1pÚvarrc  )rE   rF   rG   r5   rö   r÷   r™  rœ  ÚxbarrŒ   r‹   ra  ÚinfoÚierrW  r\  rb  Úfacr–  s                     €r6   rC   zbeta_gen.fit#  s•  ø€ ð �x‰x˜ Ó%ˆØ—‘˜( DÓ)ˆàˆ<˜6˜>ä‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ð 	�‰�˜ÔØ�‰�˜4Ô ä! $Ò(=Ó>ˆÜ! $Ò(=Ó>ˆä$ TÔ*àˆ>˜b˜näð )ó *ð *ô �{‰{˜4Ó ×$Ñ$Ô&ÜÐCÓDÐDô —‘˜“ Ñ%¨Ñ/ˆÜ�6‰6�$˜!‘)Ô¤§¡ t¨q¡yÔ 1Ü˜v¨T¸À¹ÔGÐGà�y‰y‹{ˆàˆ>˜R˜^ð ˆ~ð �Ø˜4‘x�Ø˜4‘x‘à�ð �D‘˜A ™HÑ%ˆAô &.§_¡_Ü˜QØœ˜T›¤B§F¡F¨4£L×$4Ñ$4Ó$6Ð7Ø ô&Ñ"ˆE�4˜˜dð
 �aŠxÜ$¨$Ô/Ð/Ø�a‘ˆAàˆ~ð ˜!�1‘ô —‘˜“×!Ñ!Ó#ˆBÜ—‘˜4˜%“×$Ñ$Ó&ˆBð ˜!˜d™(Ñ# d§h¡h°A hÓ&6Ñ6¸Ñ:ˆCØ�s‘
ˆAØ�T‘˜SÑ ˆAô &.§_¡_Ü˜q !˜fÜ˜$“i  RÐ(Ø ô&Ñ"ˆE�4˜˜dð
 �aŠxÜ$¨$Ô/Ð/Ø‰DˆAˆqà�!�T˜6Ð!Ð!r8   c                 óÖ   — d„ }d„ }d„ }d„ }|dk\  r|dk\  r	 |||«      S |dk  r||z
  dk\  r| ||«      k\  r	 |||«      S |dk  r||z
  dk\  r| ||«      k\  r	 |||«      S  |||«      S )Nc                 óâ   — t        j                  | |«      | dz
  t        j                  | «      z  z
  |dz
  t        j                  |«      z  z
  | |z   dz
  t        j                  | |z   «      z  z   S r1  )rw   ry  r[  ©r‹   rŒ   s     r6   Úregularz"beta_gen._entropy.<locals>.regular�  sf   € Ü—I‘I˜a “O q¨1¡u´·±°q³	Ñ&9Ñ9Ø˜‘UœbŸf™f Q›iÑ'ñ(Ø+,¨q©5°1©9¼¿¹¸qÀ1¹u»Ñ*EñFð Gr8   c                 ó¬  — | |z   }dt        j                  dt         j                  z  «      t        j                  | «      z   t        j                  |«      z   dt        j                  |«      z  z
  dz   z  }d|z  d|dz  z  z   |dz  z   d|d	z  z  z
  }d
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  | dz  z
  | d	z  z   }d
|z  d|dz  z  z
  |dz  z
  |d	z  z   }|||z   |z   dz  z   S )Nr”   rU   r†  r   én   é   ç       Àç      Àç      ÀiÎÿÿÿé
   éx   rï   )r‹   rŒ   Úsum_abÚlog_termÚt1Út2Út3s          r6   Úasymptotic_ab_largez.beta_gen._entropy.<locals>.asymptotic_ab_large“  sï   € Ø˜‘UˆFØÜ—‘�qœŸ™‘w“¤"§&¡&¨£)Ñ+¬b¯f©f°Q«iÑ7¸!¼B¿F¹FÀ6»NÑ:JÑJÈQÑNñˆHð �V‘˜b ¨¡™oÑ-°¸±Ñ<¸qÀÈÁ¹~ÑMˆBØ�Q‘˜˜A˜t™G™Ñ# a¨¡gÑ-°°4±Ñ7ˆBØ�Q‘˜˜A˜t™G™Ñ# a¨¡gÑ-°°4±Ñ7ˆBØ˜r B™w¨™|¨sÑ2Ñ2Ð2r8   c                 ó  — | |z   }t        j                  | «      | dz
  t        j                  | «      z  z
  }dd|z  z  dd|z  z  z   |dz  dz  z
  |dz  dz  z
  |dz  dz  z   |d	z  d
z  z   |dz  d
z  z
  d|z  z   dd|z  z  z
  |dz  dz  z   |dz  dz  z   |dz  dz  z
  |d	z  d
z  z
  |dz  dz  z   }|t        j                  | |z  «      z  t        j
                  |«      z   dt        j
                  |«      z  z
  }||z   |z   S )Nr   éÿÿÿÿrU   é   r³  r´  r·  rµ  ç      Àéü   ç      Àr…  é<   é~   )rw   Úgammalnr[  rP   r¦  rð   )r‹   rŒ   r¸  rº  r»  r¹  s         r6   Úasymptotic_b_largez-beta_gen._entropy.<locals>.asymptotic_b_large�  sM  € Ø˜‘UˆFÜ—‘˜A“ ! a¡%¬2¯6©6°!«9Ñ!4Ñ4ˆBà�Q�q‘S‘	˜A˜r !™t™HÑ$ q¨$¡w¨r¡zÑ1°A°t±G¸C±KÑ?À!ÀTÁ'È#Á+ÑMØ�T‘'˜#‘+ñØ ! 4¡¨¡ñ,Ø./°©hñ7Ø9:¸B¸v¹I¹ñGà˜$‘,˜q‘.ñ!à#)¨4¡<°Ñ#3ñ4à6<¸d±lÀ2±oñFð ˜$‘,˜sÑ"ñ#ð &,¨T¡\°#Ñ%5ñ6ð ð œbŸh™h q¨¡s›mÑ+¬b¯f©f°Q«iÑ7¸!¼B¿F¹FÀ6»NÑ:JÑJˆHØ˜‘7˜XÑ%Ð%r8   c                 óŒ   — | dk(  ryt        j                  | «      }t        |«      }t        | d|z  z  «      dz   }|dd|z   z  z  S )Nr‰   iè  r¶  rU   é   )rP   Úlog10r  )ÚvÚjÚdigitsÚds       r6   Úthreshold_largez*beta_gen._entropy.<locals>.threshold_large©  sL   € Ø�CŠxØä—‘˜“ˆAÜ˜“VˆFÜ�A˜˜f™Ñ$Ó%¨Ñ)ˆAØ�R˜!˜a™%‘[‘=Ð r8   g    ÀëRAg    (±RAg    €„.Ar‡   )rE   r‹   rŒ   r¯  r½  rÇ  rÏ  s          r6   rò   zbeta_gen._entropyŽ  s“   € ò	Gò	3ò
	&ò	!ð �Š;˜1 š;Ù& q¨!Ó,Ð,Ø�%ŠZ˜A ™E SšL¨Q±/À!Ó2DÒ-DÙ% a¨Ó+Ð+Ø�%ŠZ˜A ™E SšL¨Q±/À!Ó2DÒ-DÙ% a¨Ó+Ð+á˜1˜a“=Ð r8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   ry   r�   r~   r   r•  rK   r	   r   rC   rò   Ú__classcell__©r–  s   @r6   re  re  Â  sj   ø„ ñò>ó
-ò*òò
#ò$ò'ò&ò#ô 4ð" Ù˜}ð 5+ô ,ó
c"ó,ó ðc"öJ+!r8   re  rn  c                   óZ   — e Zd ZdZej
                  Zd„ Zdd„Zd„ Z	d„ Z
d„ Zd„ Zd	„ Zd
„ Zy)Úbetaprime_gena«  A beta prime continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `betaprime` is:

    .. math::

        f(x, a, b) = \frac{x^{a-1} (1+x)^{-a-b}}{\beta(a, b)}

    for :math:`x >= 0`, :math:`a > 0`, :math:`b > 0`, where
    :math:`\beta(a, b)` is the beta function (see `scipy.special.beta`).

    `betaprime` takes ``a`` and ``b`` as shape parameters.

    The distribution is related to the `beta` distribution as follows:
    If :math:`X` follows a beta distribution with parameters :math:`a, b`,
    then :math:`Y = X/(1-X)` has a beta prime distribution with
    parameters :math:`a, b` ([1]_).

    The beta prime distribution is a reparametrized version of the
    F distribution.  The beta prime distribution with shape parameters
    ``a`` and ``b`` and ``scale = s`` is equivalent to the F distribution
    with parameters ``d1 = 2*a``, ``d2 = 2*b`` and ``scale = (a/b)*s``.
    For example,

    >>> from scipy.stats import betaprime, f
    >>> x = [1, 2, 5, 10]
    >>> a = 12
    >>> b = 5
    >>> betaprime.pdf(x, a, b, scale=2)
    array([0.00541179, 0.08331299, 0.14669185, 0.03150079])
    >>> f.pdf(x, 2*a, 2*b, scale=(a/b)*2)
    array([0.00541179, 0.08331299, 0.14669185, 0.03150079])

    %(after_notes)s

    References
    ----------
    .. [1] Beta prime distribution, Wikipedia,
           https://en.wikipedia.org/wiki/Beta_prime_distribution

    %(example)s

    c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS rg  rh   rh  s      r6   rk   zbetaprime_gen._shape_infoñ  rk  r8   Nc                 ól   — t         j                  |||¬«      }t         j                  |||¬«      }||z  S ©N©r×   rØ   )ÚgammaÚrvs)rE   r‹   rŒ   r×   rØ   Úu1Úu2s          r6   rÙ   zbetaprime_gen._rvsö  s3   € Ü�Y‰Y�q˜t°,ˆYÓ?ˆÜ�Y‰Y�q˜t°,ˆYÓ?ˆØ�B‰wˆr8   c                 óN   — t        j                  | j                  |||«      «      S rN   ©rP   r·   rÞ   rt  s       r6   rr   zbetaprime_gen._pdfû  ó   € ä�v‰v�d—l‘l 1 a¨Ó+Ó,Ð,r8   c                 ó–   — t        j                  |dz
  |«      t        j                  ||z   |«      z
  t        j                  ||«      z
  S r  )rw   rx  rw  ry  rt  s       r6   rÞ   zbetaprime_gen._logpdfÿ  s:   € Ü�x‰x˜˜C™ Ó#¤b§j¡j°°Q±¸Ó&:Ñ:¼R¿Y¹YÀqÈ!»_ÑLÐLr8   c                 ó0   — t        |dkD  |||gd„ d„ ¬«      S )Nr   c                 ó<   — t         j                  dd| z   z  ||«      S r^   ©rn  ry   ©Úx_Úa_Úb_s      r6   ú<lambda>z$betaprime_gen._cdf.<locals>.<lambda>  s   € œtŸx™x¨¨1¨R©4©°"°bÓ9€ r8   c                 ó<   — t         j                  | d| z   z  ||«      S r^   ©rn  ru   rã  s      r6   rç  z$betaprime_gen._cdf.<locals>.<lambda>  s   € ¤$§)¡)¨B°°"±©I°r¸2Ó">€ r8   ©Úf2©r   rt  s       r6   ru   zbetaprime_gen._cdf  s(   € ô Ø�‰E�A�q˜!�9Ù9Ù>ô@ð 	@r8   c                 ó0   — t        |dkD  |||gd„ d„ ¬«      S )Nr   c                 ó<   — t         j                  dd| z   z  ||«      S r^   ré  rã  s      r6   rç  z#betaprime_gen._sf.<locals>.<lambda>  s   € œtŸy™y¨¨A¨b©D©°2°rÓ:€ r8   c                 ó<   — t         j                  | d| z   z  ||«      S r^   râ  rã  s      r6   rç  z#betaprime_gen._sf.<locals>.<lambda>  s   € ¤$§(¡(¨2¨q°©t©9°b¸"Ó"=€ r8   rê  rì  rt  s       r6   ry   zbetaprime_gen._sf  s$   € ÜØ�‰E�A�q˜!�9Ù:Ù=ô
ð 	
r8   c                 óÔ  — t        j                  |||«      \  }}}t        j                  j	                  |||«      }t        j
                  d¬«      5  |d|z
  z  }d d d «       |dkD  }t        j                  |«      r+|r'dt        j                  j                  |||«      z  dz
  }S dt        j                  j                  ||   ||   ||   «      z  dz
  |<   |S # 1 sw Y   ŒƒxY w)Nr9  r:  r   g§èH.ÿï?)rP   Úbroadcast_arraysÚstatsrn  r~   r<  Úisscalarr�   )rE   Úpr‹   rŒ   ÚrÚoutÚrnear1s          r6   r~   zbetaprime_gen._ppf  s×   € Ü×%Ñ% a¨¨AÓ.‰ˆˆ1ˆaô �J‰J�O‰O˜A˜q !Ó$ˆÜ�[‰[ Ô)ñ 	Ø�q˜1‘u‘+ˆC÷	à�V‘ˆÜ�;‰;�qŒ>ÙØœŸ
™
Ÿ™¨¨1¨aÓ0Ñ0°1Ñ4�ð ˆ
ð œEŸJ™JŸO™O¨A¨f©I°q¸±yÀ!ÀFÁ)ÓLÑLÈqÑPˆC�‰KØˆ
÷	ð 	ús   Á	CÃC'c                 óN   ‡— t        |‰kD  ||fˆfd„t        j                  ¬«      S )Nc                 óœ   •— t        j                  t        dt        ‰«      dz   «      D �cg c]  }| |z   dz
  ||z
  z  ‘Œ c}d¬«      S c c}w )Nr   r   ©Úaxis)rP   ÚprodÚranger  )r‹   rŒ   Úirb   s      €r6   rç  z%betaprime_gen._munp.<locals>.<lambda>+  s@   ø€ œŸ™¼¸qÄ#ÀaÃ&ÈÁ(Ó9KÖ!L°A 1 Q¡3 q¡5¨1¨Q©3£-Ò!LÐSTÔU€ ùÒ!Ls   «A	©Ú	fillvalue©r   rP   ri   )rE   rb   r‹   rŒ   s    `  r6   r  zbetaprime_gen._munp(  s'   ø€ ÜØ�‰E�A�q�6ÛUÜ—f‘fôð 	r8   r  )rƒ   r„   r…   r†   r   r  r  rk   rÙ   rr   rÞ   ru   ry   r~   r  r‡   r8   r6   rÓ  rÓ  ¿  s?   „ ñ.ð^ "×4Ñ4€Mòó
ò
-òMò@ò
òó$r8   rÓ  Ú	betaprimec                   ó6   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd	d„Zd„ Z	y)
Úbradford_genab  A Bradford continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `bradford` is:

    .. math::

        f(x, c) = \frac{c}{\log(1+c) (1+cx)}

    for :math:`0 <= x <= 1` and :math:`c > 0`.

    `bradford` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS ©NÚcFr   r
  rh   rj   s    r6   rk   zbradford_gen._shape_infoH  r  r8   c                 óD   — |||z  dz   z  t        j                  |«      z  S r  ©rw   r¦  ©rE   rq   r  s      r6   rr   zbradford_gen._pdfK  s!   € à�A�a‘C˜#‘I‰¤§¡¨!£Ñ,Ð,r8   c                 ó^   — t        j                  ||z  «      t        j                  |«      z  S rN   r	  r
  s      r6   ru   zbradford_gen._cdfO  s!   € Ü�x‰x˜˜!™‹}œrŸx™x¨›{Ñ*Ð*r8   c                 ó^   — t        j                  |t        j                  |«      z  «      |z  S rN   ©rw   Úexpm1r¦  ©rE   r}   r  s      r6   r~   zbradford_gen._ppfR  s"   € Ü�x‰x˜œBŸH™H Q›K™Ó(¨1Ñ,Ð,r8   c                 ój  — t        j                  d|z   «      }||z
  ||z  z  }|dz   |z  d|z  z
  d|z  |z  |z  z  }d }d }d|v r{t        j                  d«      d|z  |z  d|z  |z  |dz   z  z
  d|z  |z  ||dz   z  dz   z  z   z  }|t        j                  |||dz
  z  d|z  z   z  «      d|z  |dz
  z  d|z  z   z  z  }d	|v rj|dz  |dz
  z  |d|z  d
z
  z  dz   z  d|z  |z  |z  |dz
  z  |dz
  z  z   d|z  |z  |z  d|z  dz
  z  z   d|dz  z  z   }|d|z  ||dz
  z  d|z  z   dz  z  z  }||||fS )Nr‰   r¶   rU   ÚsrÀ  é	   r†  r…  Úkr,  é   r$  é   )rP   rð   rÿ   )rE   r  Úmomentsr  rC  rD  rE  rF  s           r6   r   zbradford_gen._statsU  sž  € Ü�F‰F�3�q‘5‹MˆØ�‰c�A�a‘C‰[ˆØ�#‘�q‰y˜˜Q™‰  1¡ Q¡ q¡Ñ)ˆØˆØˆØ�'‰>Ü—‘˜“˜R ™T !™V A a¡C¨¡E¨1¨Q©3¡KÑ/°°!±°A±°q¸!¸A¹#±w¸q±yÑ0AÑAÑBˆBØ”"—'‘'˜!˜Q  !¡™W Q q¡S™[™/Ó*¨A¨a©C°°1±©I°a¸±c©MÑ:Ñ:ˆBØ�'‰>Ø�Q‘$˜˜!™‘*˜a  1¡ R¡™j¨™mÑ,¨R°©T°!©V°A©X°q¸±s©^¸Q¸q¹SÑ-AÑAØ�A‘#�a‘%˜‘'˜1˜Q™3˜r™6Ñ"ñ#Ø%'¨¨1©¡Wñ-ˆBà�!�A‘#�q˜!˜A™#‘w˜q ™s‘{ QÑ&Ñ&Ñ&ˆBØ�3˜˜BˆÐr8   c                 ón   — t        j                  d|z   «      }|dz  t        j                  ||z  «      z
  S ©Nr   r¶   r2  )rE   r  r  s      r6   rò   zbradford_gen._entropyd  s.   € Ü�F‰F�1�Q‘3‹KˆØ�‰u”r—v‘v˜a ™c“{Ñ"Ð"r8   N©ÚmvrH  r‡   r8   r6   r  r  2  s&   „ ñò*Eò-ò+ò-óó#r8   r  Úbradfordc                   óR   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zd„ Zy)Úburr_genaˆ  A Burr (Type III) continuous random variable.

    %(before_notes)s

    See Also
    --------
    fisk : a special case of either `burr` or `burr12` with ``d=1``
    burr12 : Burr Type XII distribution
    mielke : Mielke Beta-Kappa / Dagum distribution

    Notes
    -----
    The probability density function for `burr` is:

    .. math::

        f(x; c, d) = c d \frac{x^{-c - 1}}
                              {{(1 + x^{-c})}^{d + 1}}

    for :math:`x >= 0` and :math:`c, d > 0`.

    `burr` takes ``c`` and ``d`` as shape parameters for :math:`c` and
    :math:`d`.

    This is the PDF corresponding to the third CDF given in Burr's list;
    specifically, it is equation (11) in Burr's paper [1]_. The distribution
    is also commonly referred to as the Dagum distribution [2]_. If the
    parameter :math:`c < 1` then the mean of the distribution does not
    exist and if :math:`c < 2` the variance does not exist [2]_.
    The PDF is finite at the left endpoint :math:`x = 0` if :math:`c * d >= 1`.

    %(after_notes)s

    References
    ----------
    .. [1] Burr, I. W. "Cumulative frequency functions", Annals of
       Mathematical Statistics, 13(2), pp 215-232 (1942).
    .. [2] https://en.wikipedia.org/wiki/Dagum_distribution
    .. [3] Kleiber, Christian. "A guide to the Dagum distributions."
       Modeling Income Distributions and Lorenz Curves  pp 97-117 (2008).

    %(example)s

    c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS ©Nr  Fr   r
  rÎ  rh   ©rE   ÚicÚids      r6   rk   zburr_gen._shape_info�  rk  r8   c                 ó\   — t        |dk(  |||gd„ d„ ¬«      }|j                  dk(  r|d   S |S )Nr   c                 ó6   — ||z  | ||z  dz
  z  z  d| |z  z   z  S r^   r‡   ©rä  Úc_Úd_s      r6   rç  zburr_gen._pdf.<locals>.<lambda>¦  s(   € ˜r B™w¨"¨r°"©u°Q©w©-Ñ8¸AÀÀBÁ¹JÑG€ r8   c                 ó@   — ||z  | | dz
  z  z  d| | z  z   |dz   z  z  S ©Nr‰   r   r‡   r%  s      r6   rç  zburr_gen._pdf.<locals>.<lambda>§  s6   €  2¨¡7¨b°b°S¸3±YÑ.?Ñ#@Ø%&¨°°©¡_¸"¸s¹(Ñ$Cñ$E€ r8   rê  r‡   ©r   Úndim©rE   rq   r  rÎ  Úoutputs        r6   rr   zburr_gen._pdf¢  sB   € äØ�‰F�Q˜˜1�IÙGñFôGˆð
 �;‰;˜!ÒØ˜"‘:ÐØˆr8   c                 ó\   — t        |dk(  |||gd„ d„ ¬«      }|j                  dk(  r|d   S |S )Nr   c                 óÐ   — t        j                  |«      t        j                  |«      z   t        j                  ||z  dz
  | «      z   |dz   t        j                  | |z  «      z  z
  S r^   )rP   rð   rw   rx  r¦  r%  s      r6   rç  z"burr_gen._logpdf.<locals>.<lambda>°  sQ   € ¤§¡ r£
¬R¯V©V°B«ZÑ 7¼"¿(¹(À2ÀbÁ5È1Á9ÈbÓ:QÑ QØ#% a¡4¬2¯8©8°B¸±HÓ+=Ñ"=ñ!>€ r8   c                 óÊ   — t        j                  |«      t        j                  |«      z   t        j                  | dz
  | «      z   t        j                  |dz   | | z  «      z
  S r^   ©rP   rð   rw   rx  rw  r%  s      r6   rç  z"burr_gen._logpdf.<locals>.<lambda>²  sR   € ¤2§6¡6¨"£:´·±°r³
Ñ#:Ü%'§X¡X¨r¨c°A©g°rÓ%:ñ$;ä%'§Z¡Z°°1±°b¸B¸3±iÓ%@ñ$A€ r8   rê  r‡   r*  r,  s        r6   rÞ   zburr_gen._logpdf­  sD   € ÜØ�‰F�Q˜˜1�Iñ?ñBô	Cˆð �;‰;˜!ÒØ˜"‘:ÐØˆr8   c                 ó   — d|| z  z   | z  S r^   r‡   ©rE   rq   r  rÎ  s       r6   ru   zburr_gen._cdf¹  s   € Ø�A˜˜‘G‘ ˜rÑ"Ð"r8   c                 ó<   — t        j                  || z  «      | z  S rN   r	  r3  s       r6   rã   zburr_gen._logcdf¼  s   € Ü�x‰x˜˜Q˜B™Ó  Q BÑ'Ð'r8   c                 óN   — t        j                  | j                  |||«      «      S rN   ©rP   r·   rç   r3  s       r6   ry   zburr_gen._sf¿  ó   € Ü�v‰v�d—k‘k ! Q¨Ó*Ó+Ð+r8   c                 óD   — t        j                  d|| z  z   | z   «      S r^   ©rP   r¦  r3  s       r6   rç   zburr_gen._logsfÂ  s%   € Ü�x‰x˜1˜q A 2™w™;¨1¨"Ñ-Ð-Ó.Ð.r8   c                 ó$   — |d|z  z  dz
  d|z  z  S ©Nç      ð¿r   r‡   ©rE   r}   r  rÎ  s       r6   r~   zburr_gen._ppfÅ  s   € Ø�D˜‘F‘˜a‘ 4¨¡6Ñ*Ð*r8   c                 ól   — t        j                  d|z  | «      }t        j                  |«      d|z  z  S ©Nr<  ©rw   rw  r  )rE   r}   r  rÎ  Ú_qs        r6   r�   zburr_gen._isfÈ  s/   € Ü�Z‰Z˜˜q™ 1 "Ó%ˆÜ�x‰x˜‹|  q¡Ñ)Ð)r8   c           	      óž  — t        j                  dd«      j                  dd«      |z  }t        j                  ||z   d|z
  «      |z  \  }}}}t        j
                  |dkD  |t         j                  «      }||dz  z
  }	t        j
                  |dkD  |	t         j                  «      }
t        |dkD  |||||	fd„ t         j                  ¬	«      }t        |d
kD  ||||||	fd„ t         j                  ¬	«      }t        j                  |«      dk(  r>|j                  «       |
j                  «       |j                  «       |j                  «       fS ||
||fS )Nr   é   r$  r‰   rU   r¶   ç      @c                 ó\   — |d|z  |z  z
  d|dz  z  z   t        j                  |dz  «      z  S )Nr†  rU   r‡  )r  Úe1Úe2Úe3Úmu2_if_cs        r6   rç  z!burr_gen._stats.<locals>.<lambda>Ö  s5   € ¨b°1°R±4¸±7©l¸Q¸rÀ1¹u¹WÑ.DÜ/1¯w©w¸À1±}Ó/Eñ.F€ r8   rÿ  ç      @c                 óT   — |d|z  |z  z
  d|z  |dz  z  z   d|dz  z  z
  |dz  z  dz
  S )Nr$  r…  rU   r†  r‡   )r  rF  rG  rH  Úe4rI  s         r6   rç  z!burr_gen._stats.<locals>.<lambda>Ü  sB   € Ø�q˜‘t˜B‘w‘,  2¡ b¨!¡e¡Ñ+¨a°°A±©gÑ5¸À1¹ÑDÈÑIð r8   r   )
rP   ÚarangeÚreshaperw   rn  Úwherer  r   r+  Úitem)rE   r  rÎ  ÚncrF  rG  rH  rL  rC  rI  rD  rE  rF  s                r6   r   zburr_gen._statsÌ  s+  € Ü�Y‰Y�q˜!‹_×$Ñ$ Q qÓ)¨AÑ-ˆäŸ™  R¡¨¨b©Ó1°AÑ5‰ˆˆB��BÜ�X‰X�a˜#‘g˜r¤2§6¡6Ó*ˆØ˜˜A™‘:ˆÜ�h‰h�q˜3‘w ¬"¯&©&Ó1ˆÜØ�‰GØ��B˜˜HÐ%ñGä—f‘fôˆô Ø�‰GØ��B˜˜B Ð)ñKä—f‘fôˆô �7‰7�1‹:˜Š?Ø—7‘7“9˜cŸh™h›j¨"¯'©'«)°R·W±W³YÐ>Ð>Ø�3˜˜BˆÐr8   c                 óê   ‡— d„ Št        j                  |«      t        j                  |«      t        j                  |«      }}}t        ||kD  ||k(  z  ||k(  z  |||fˆfd„t         j                  «      S )Nc                 óP   — d| z  |z  }|t        j                  d|z
  ||z   «      z  S r  ©rw   rn  ©rb   r  rÎ  rQ  s       r6   Ú__munpzburr_gen._munp.<locals>.__munpä  ó-   € Ø�a‘˜!‘ˆBØ”r—w‘w˜s R™x¨¨R©Ó0Ñ0Ð0r8   c                 ó   •—  ‰|| |«      S rN   r‡   )r  rÎ  rb   Ú_burr_gen__munps      €r6   rç  z burr_gen._munp.<locals>.<lambda>é  s   ø€ ©&°°A°q«/€ r8   )rP   rû   r   r  )rE   rb   r  rÎ  rY  s       @r6   r  zburr_gen._munpã  sf   ø€ ò	1ô —*‘*˜Q“-¤§¡¨A£´·
±
¸1³ˆaˆ1ˆÜ˜1˜q™5 Q¨!¡VÑ,°°Q±Ñ7¸!¸QÀ¸Û9ÜŸ&™&ó"ð 	"r8   N)rƒ   r„   r…   r†   rk   rr   rÞ   ru   rã   ry   rç   r~   r�   r   r  r‡   r8   r6   r  r  l  s?   „ ñ+ò`ò
	ò
ò#ò(ò,ò/ò+ò*òó."r8   r  Úburrc                   óL   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zy)Ú
burr12_gena}  A Burr (Type XII) continuous random variable.

    %(before_notes)s

    See Also
    --------
    fisk : a special case of either `burr` or `burr12` with ``d=1``
    burr : Burr Type III distribution

    Notes
    -----
    The probability density function for `burr12` is:

    .. math::

        f(x; c, d) = c d \frac{x^{c-1}}
                              {(1 + x^c)^{d + 1}}

    for :math:`x >= 0` and :math:`c, d > 0`.

    `burr12` takes ``c`` and ``d`` as shape parameters for :math:`c`
    and :math:`d`.

    This is the PDF corresponding to the twelfth CDF given in Burr's list;
    specifically, it is equation (20) in Burr's paper [1]_.

    %(after_notes)s

    The Burr type 12 distribution is also sometimes referred to as
    the Singh-Maddala distribution from NIST [2]_.

    References
    ----------
    .. [1] Burr, I. W. "Cumulative frequency functions", Annals of
       Mathematical Statistics, 13(2), pp 215-232 (1942).

    .. [2] https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm

    .. [3] "Burr distribution",
       https://en.wikipedia.org/wiki/Burr_distribution

    %(example)s

    c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS r  rh   r   s      r6   rk   zburr12_gen._shape_info  rk  r8   c                 óN   — t        j                  | j                  |||«      «      S rN   rÝ  r3  s       r6   rr   zburr12_gen._pdf"  rÞ  r8   c                 óÈ   — t        j                  |«      t        j                  |«      z   t        j                  |dz
  |«      z   t        j                  | dz
  ||z  «      z   S r^   r1  r3  s       r6   rÞ   zburr12_gen._logpdf&  sL   € Ü�v‰v�a‹yœ2Ÿ6™6 !›9Ñ$¤r§x¡x°°A±°qÓ'9Ñ9¼B¿J¹JÈÀrÈ!ÁtÈQÐPQÉTÓ<RÑRÐRr8   c                 óP   — t        j                  | j                  |||«      «       S rN   ©rw   r  rç   r3  s       r6   ru   zburr12_gen._cdf)  ó!   € Ü—‘˜Ÿ™ Q¨¨1Ó-Ó.Ð.Ð.r8   c                 óB   — t        j                  d||z  z   | z   «      S r^   r	  r3  s       r6   rã   zburr12_gen._logcdf,  s#   € Ü�x‰x˜!˜a ™d™( q bÑ)Ð)Ó*Ð*r8   c                 óN   — t        j                  | j                  |||«      «      S rN   r6  r3  s       r6   ry   zburr12_gen._sf/  r7  r8   c                 ó6   — t        j                  | ||z  «      S rN   ©rw   rw  r3  s       r6   rç   zburr12_gen._logsf2  s   € Ü�z‰z˜1˜"˜a ™dÓ#Ð#r8   c                 ól   — t        j                  d|z  t        j                  | «      z  «      d|z  z  S ©Nr¿  r   r  r=  s       r6   r~   zburr12_gen._ppf5  s/   € ô �x‰x˜˜1™œrŸx™x¨¨›|Ñ+Ó,¨q°©sÑ3Ð3r8   c                 ój   — t        j                  d|z  t        j                  |«      z  «      d|z  z  S rh  )rw   r  rP   rð   )rE   rô  r  rÎ  s       r6   r�   zburr12_gen._isf;  s+   € Ü�x‰x˜˜1™œrŸv™v a›yÑ(Ó)¨A¨a©CÑ0Ð0r8   c                 óT   — d„ }t        ||z  |kD  |||f|t        j                  ¬«      S )Nc                 óP   — d| z  |z  }|t        j                  d|z   ||z
  «      z  S r  rT  rU  s       r6   Úmoment_if_existsz*burr12_gen._munp.<locals>.moment_if_exists?  rW  r8   rÿ  ©r   rP   r  )rE   rb   r  rÎ  rl  s        r6   r  zburr12_gen._munp>  s2   € ò	1ô ˜!˜a™% !™) a¨¨A YÐ0@Ü$&§F¡Fô,ð 	,r8   N)rƒ   r„   r…   r†   rk   rr   rÞ   ru   rã   ry   rç   r~   r�   r  r‡   r8   r6   r\  r\  ð  s;   „ ñ+òXò
-òSò/ò+ò,ò$ò4ò1ó,r8   r\  Úburr12c                   óX   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zd„ Zd„ Zy)Úfisk_genaº  A Fisk continuous random variable.

    The Fisk distribution is also known as the log-logistic distribution.

    %(before_notes)s

    See Also
    --------
    burr

    Notes
    -----
    The probability density function for `fisk` is:

    .. math::

        f(x, c) = \frac{c x^{c-1}}
                       {(1 + x^c)^2}

    for :math:`x >= 0` and :math:`c > 0`.

    Please note that the above expression can be transformed into the following
    one, which is also commonly used:

    .. math::

        f(x, c) = \frac{c x^{-c-1}}
                       {(1 + x^{-c})^2}

    `fisk` takes ``c`` as a shape parameter for :math:`c`.

    `fisk` is a special case of `burr` or `burr12` with ``d=1``.

    Suppose ``X`` is a logistic random variable with location ``l``
    and scale ``s``. Then ``Y = exp(X)`` is a Fisk (log-logistic)
    random variable with ``scale = exp(l)`` and shape ``c = 1/s``.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zfisk_gen._shape_infou  r  r8   c                 ó0   — t         j                  ||d«      S r  )rZ  rr   r
  s      r6   rr   zfisk_gen._pdfx  s   € ä�y‰y˜˜A˜sÓ#Ð#r8   c                 ó0   — t         j                  ||d«      S r  )rZ  ru   r
  s      r6   ru   zfisk_gen._cdf|  ó   € Ü�y‰y˜˜A˜sÓ#Ð#r8   c                 ó0   — t         j                  ||d«      S r  )rZ  ry   r
  s      r6   ry   zfisk_gen._sf  s   € Ü�x‰x˜˜1˜cÓ"Ð"r8   c                 ó0   — t         j                  ||d«      S r  )rZ  rÞ   r
  s      r6   rÞ   zfisk_gen._logpdf‚  s   € ä�|‰|˜A˜q #Ó&Ð&r8   c                 ó0   — t         j                  ||d«      S r  )rZ  rã   r
  s      r6   rã   zfisk_gen._logcdf†  s   € Ü�|‰|˜A˜q #Ó&Ð&r8   c                 ó0   — t         j                  ||d«      S r  )rZ  rç   r
  s      r6   rç   zfisk_gen._logsf‰  s   € Ü�{‰{˜1˜a Ó%Ð%r8   c                 ó0   — t         j                  ||d«      S r  )rZ  r~   r
  s      r6   r~   zfisk_gen._ppfŒ  rt  r8   c                 ó0   — t         j                  ||d«      S r  )rZ  r�   r  s      r6   r�   zfisk_gen._isf�  rt  r8   c                 ó0   — t         j                  ||d«      S r  )rZ  r  ©rE   rb   r  s      r6   r  zfisk_gen._munp’  s   € Ü�z‰z˜!˜Q Ó$Ð$r8   c                 ó.   — t         j                  |d«      S r  )rZ  r   ©rE   r  s     r6   r   zfisk_gen._stats•  s   € Ü�{‰{˜1˜cÓ"Ð"r8   c                 ó2   — dt        j                  |«      z
  S r  r2  r~  s     r6   rò   zfisk_gen._entropy˜  ó   € Ø”2—6‘6˜!“9‰}Ðr8   N)rƒ   r„   r…   r†   rk   rr   ru   ry   rÞ   rã   rç   r~   r�   r  r   rò   r‡   r8   r6   rp  rp  J  sE   „ ñ)òTEò$ò$ò#ò'ò'ò&ò$ò$ò%ò#ór8   rp  Úfiskc                   óN   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zdd„Zy)Ú
cauchy_genaý  A Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `cauchy` is

    .. math::

        f(x) = \frac{1}{\pi (1 + x^2)}

    for a real number :math:`x`.

    This distribution uses routines from the Boost Math C++ library for
    the computation of the ``ppf` and ``isf`` methods. [1]_

    %(after_notes)s

    References
    ----------
    .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zcauchy_gen._shape_infoº  r¦   r8   c                 ó’   — t        j                  d¬«      5  dt         j                  z  d||z  z   z  cd d d «       S # 1 sw Y   y xY w)Nr9  rq  r‰   )rP   r<  rñ   r©   s     r6   rr   zcauchy_gen._pdf½  s;   € ä�[‰[˜hÔ'ñ 	'Ø”r—u‘u‘9˜c ! A¡#™gÑ&÷	'÷ 	'ò 	'ús	   —=½Ac                 óZ   — t        j                  |«      }t        |dk  |fd„ d„ ¬«      }|S )Nr   c                 óB   — t          t        j                  | dz  «      z
  S r  )r(   rP   r¦  ©Úabsxs    r6   rç  z$cauchy_gen._logpdf.<locals>.<lambda>Í  s   € ¤w h´·±¸$À¹'Ó1BÑ&B€ r8   c                 óz   — t          dt        j                  | «      z  t        j                  d| z  dz  «      z   z
  S ©NrU   r   )r(   rP   rð   r¦  rˆ  s    r6   rç  z$cauchy_gen._logpdf.<locals>.<lambda>Î  s3   € ¬¨Ø)*¬2¯6©6°$«<©¼"¿(¹(ÀAÀdÁFÈQÁ;Ó:OÑ)Oñ)Q€ r8   ©Úfrë  )rP   Úabsr   )rE   rq   r‰  Úys       r6   rÞ   zcauchy_gen._logpdfÂ  s8   € ô �v‰v�a‹yˆô �t˜a‘x $ ÙBñRôSˆð ˆr8   c                 óR   — t        j                  d| «      t         j                  z  S r^   ©rP   Úarctan2rñ   r©   s     r6   ru   zcauchy_gen._cdfÒ  s   € Ü�z‰z˜!˜a˜RÓ ¤§¡Ñ&Ð&r8   c                 ó0   — t        j                  |dd«      S ©Nr   r   )rn   Ú_cauchy_ppfr°   s     r6   r~   zcauchy_gen._ppfÕ  ó   € Ü�‰˜q ! QÓ'Ð'r8   c                 óP   — t        j                  d|«      t         j                  z  S r^   r‘  r©   s     r6   ry   zcauchy_gen._sfØ  s   € Ü�z‰z˜!˜QÓ¤§¡Ñ%Ð%r8   c                 ó0   — t        j                  |dd«      S r”  )rn   Ú_cauchy_isfr°   s     r6   r�   zcauchy_gen._isfÛ  r–  r8   c                 ó~   — t         j                  t         j                  t         j                  t         j                  fS rN   ©rP   r  rj   s    r6   r   zcauchy_gen._statsÞ  ó!   € Ü�v‰v”r—v‘vœrŸv™v¤r§v¡vÐ-Ð-r8   c                 óN   — t        j                  dt         j                  z  «      S r)  rï   rj   s    r6   rò   zcauchy_gen._entropyá  ó   € Ü�v‰v�aœŸ™‘g‹Ðr8   Nc                 óŽ   — t        |t        «      r|j                  «       }t        j                  |g d¢«      \  }}}|||z
  dz  fS ©N©é   é2   éK   rU   ©r?   r*   r“  rP   Ú
percentile©rE   rF   rG   Úp25Úp50Úp75s         r6   r•  zcauchy_gen._fitstartä  óA   € ä�dœLÔ)Ø—>‘>Ó#ˆDÜŸ™ dªLÓ9‰ˆˆS�#Ø�S˜3‘Y ‘MÐ!Ð!r8   rN   )rƒ   r„   r…   r†   rk   rr   rÞ   ru   r~   ry   r�   r   rò   r•  r‡   r8   r6   rƒ  rƒ  Ÿ  s9   „ ñò4ò'ò
ò 'ò(ò&ò(ò.òô"r8   rƒ  Úcauchyc                   óN   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Zd„ Zy)Úchi_genaß  A chi continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `chi` is:

    .. math::

        f(x, k) = \frac{1}{2^{k/2-1} \Gamma \left( k/2 \right)}
                   x^{k-1} \exp \left( -x^2/2 \right)

    for :math:`x >= 0` and :math:`k > 0` (degrees of freedom, denoted ``df``
    in the implementation). :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    Special cases of `chi` are:

        - ``chi(1, loc, scale)`` is equivalent to `halfnorm`
        - ``chi(2, 0, scale)`` is equivalent to `rayleigh`
        - ``chi(3, 0, scale)`` is equivalent to `maxwell`

    `chi` takes ``df`` as a shape parameter.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS ©NÚdfFr   r
  rh   rj   s    r6   rk   zchi_gen._shape_info  ó   € Ü˜4 ¨¬B¯F©F¨°^ÓDÐEÐEr8   Nc                 óX   — t        j                  t        j                  |||¬«      «      S rÖ  )rP   rÿ   Úchi2rÙ  ©rE   r±  r×   rØ   s       r6   rÙ   zchi_gen._rvs  s    € Ü�w‰w”t—x‘x ¨¸L�xÓIÓJÐJr8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  ©rE   rq   r±  s      r6   rr   zchi_gen._pdf  s   € ô �v‰v�d—l‘l 1 bÓ)Ó*Ð*r8   c                 óà   — t        j                  d«      dt        j                  d«      z  |z  z
  t        j                  d|z  «      z
  }|t        j                  |dz
  |«      z   d|dz  z  z
  S )NrU   r”   r‰   )rP   rð   rw   rÆ  rx  )rE   rq   r±  Úls       r6   rÞ   zchi_gen._logpdf  s]   € Ü�F‰F�1‹I˜œ2Ÿ6™6 !›9™ R™Ñ'¬"¯*©*°R¸±UÓ*;Ñ;ˆØ”2—8‘8˜B ™G QÓ'Ñ'¨"¨Q°©T©'Ñ1Ð1r8   c                 ó@   — t        j                  d|z  d|dz  z  «      S ©Nr”   rU   ©rw   Úgammaincr·  s      r6   ru   zchi_gen._cdf  s   € Ü�{‰{˜2˜b™5 " Q¨¡T¡'Ó*Ð*r8   c                 ó@   — t        j                  d|z  d|dz  z  «      S r»  ©rw   Ú	gammainccr·  s      r6   ry   zchi_gen._sf!  s   € Ü�|‰|˜B˜r™E 2 a¨¡d¡7Ó+Ð+r8   c                 ó`   — t        j                  dt        j                  d|z  |«      z  «      S ©NrU   r”   ©rP   rÿ   rw   Úgammaincinv©rE   r}   r±  s      r6   r~   zchi_gen._ppf$  s%   € Ü�w‰w�qœŸ™¨¨2©¨qÓ1Ñ1Ó2Ð2r8   c                 ó`   — t        j                  dt        j                  d|z  |«      z  «      S rÂ  ©rP   rÿ   rw   ÚgammainccinvrÅ  s      r6   r�   zchi_gen._isf'  s%   € Ü�w‰w�qœŸ™¨¨B©°Ó2Ñ2Ó3Ð3r8   c                 óz  — t        j                  d«      t        j                  d|z  d«      z  }|||z  z
  }d|dz  z  |dd|z  z
  z  z   t        j                  t        j
                  |d«      «      z  }d|z  d|z
  z  d|dz  z  z
  d|dz  z  d|z  dz
  z  z   }|t        j                  |d	z  «      z  }||||fS )
NrU   r”   rD  r   ç      ø?r‰   r…  r$  r¶   )rP   rÿ   rw   Úpochrû   Úpower©rE   r±  rC  rD  rE  rF  s         r6   r   zchi_gen._stats*  sÄ   € ä�W‰W�Q‹Zœ"Ÿ'™' #¨¡(¨CÓ0Ñ0ˆØ�2�b‘5‰jˆØ��C‘‰i˜"˜a  "¡™f™+Ñ%¤r§z¡z´"·(±(¸3ÀÓ2DÓ'EÑEˆØˆr‰T�3�r‘6‰]˜1˜R ™U™7Ñ" Q r¨1¡u¡W°°"±°Q±Ñ%7Ñ7ˆØ
Œb�j‰j˜˜c™Ó"Ñ"ˆØ�3˜˜BˆÐr8   c                 ó4   — d„ }d„ }t        |dk  |f||¬«      S )Nc                 ó¨   — t        j                  d| z  «      d| t        j                  d«      z
  | dz
  t        j                  d| z  «      z  z
  z  z   S rî   )rw   rÆ  rP   rð   Údigamma©r±  s    r6   Úregular_formulaz)chi_gen._entropy.<locals>.regular_formula5  sM   € Ü—J‘J˜r B™wÓ'Ø˜R¤"§&¡&¨£)™^¨r°A©v¼¿¹ÀCÈ"ÁHÓ9MÑ.MÑMÑNñOð Pr8   c                 óœ   — dt        j                  t         j                  «      dz  z   | dz  dz  z
  | dz  dz  z
  d| dz  z  z
  | dz  d	z  z   S )
Nr”   rU   r¿  r…  r  glÁlÁ¶?éýÿÿÿéüÿÿÿé   rï   rÑ  s    r6   Úasymptotic_formulaz,chi_gen._entropy.<locals>.asymptotic_formula9  sY   € Øœ"Ÿ&™&¤§¡›-¨™/Ñ)¨R°©V°Q©JÑ6¸"¸b¹&À!¹ÑCØ˜B ™F‘mñ$Ø')¨2¡v¨r¡kñ2ð 3r8   g     Àr@rê  rì  )rE   r±  rÒ  r×  s       r6   rò   zchi_gen._entropy3  s+   € ò	Pò	3ô ˜"˜s™( R F¨OØ/ô1ð 	1r8   r  ©rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   ry   r~   r�   r   rò   r‡   r8   r6   r®  r®  ï  s;   „ ñò<FóKò+ò2ò+ò,ò3ò4òó1r8   r®  Úchic                   óN   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Zd„ Zy)Úchi2_genaÌ  A chi-squared continuous random variable.

    For the noncentral chi-square distribution, see `ncx2`.

    %(before_notes)s

    See Also
    --------
    ncx2

    Notes
    -----
    The probability density function for `chi2` is:

    .. math::

        f(x, k) = \frac{1}{2^{k/2} \Gamma \left( k/2 \right)}
                   x^{k/2-1} \exp \left( -x/2 \right)

    for :math:`x > 0`  and :math:`k > 0` (degrees of freedom, denoted ``df``
    in the implementation).

    `chi2` takes ``df`` as a shape parameter.

    The chi-squared distribution is a special case of the gamma
    distribution, with gamma parameters ``a = df/2``, ``loc = 0`` and
    ``scale = 2``.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r°  rh   rj   s    r6   rk   zchi2_gen._shape_infof  r²  r8   Nc                 ó&   — |j                  ||«      S rN   )Ú	chisquarerµ  s       r6   rÙ   zchi2_gen._rvsi  s   € Ø×%Ñ% b¨$Ó/Ð/r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  r·  s      r6   rr   zchi2_gen._pdfl  s   € ä�v‰v�d—l‘l 1 bÓ)Ó*Ð*r8   c                 ó°   — t        j                  |dz  dz
  |«      |dz  z
  t        j                  |dz  «      z
  t        j                  d«      |z  dz  z
  S )Nr¶   r   rU   )rw   rx  rÆ  rP   rð   r·  s      r6   rÞ   zchi2_gen._logpdfp  sM   € Ü�x‰x˜˜2™˜a™ Ó# a¨¡dÑ*¬R¯Z©Z¸¸2¹Ó->Ñ>Ä"Ç&Á&ÈÃ)ÈBÁ,ÐPRÑARÑRÐRr8   c                 ó.   — t        j                  ||«      S rN   )rw   Úchdtrr·  s      r6   ru   zchi2_gen._cdfs  ó   € Ü�x‰x˜˜A‹Ðr8   c                 ó.   — t        j                  ||«      S rN   )rw   Úchdtrcr·  s      r6   ry   zchi2_gen._sfv  ó   € Ü�y‰y˜˜QÓÐr8   c                 ó.   — t        j                  ||«      S rN   )rw   Úchdtri©rE   rô  r±  s      r6   r�   zchi2_gen._isfy  ræ  r8   c                 ó:   — dt        j                  |dz  |«      z  S r  ©rw   rÄ  ré  s      r6   r~   zchi2_gen._ppf|  s   € Ø”—‘  1¡ aÓ(Ñ(Ð(r8   c                 ó\   — |}d|z  }dt        j                  d|z  «      z  }d|z  }||||fS )NrU   r¶   ç      (@r‡  rÍ  s         r6   r   zchi2_gen._stats  s=   € ØˆØ�‰dˆØŒr�w‰w�s˜2‘v‹ÑˆØ�"‰WˆØ�3˜˜BˆÐr8   c                 ó>   — d|z  }d„ }d„ }t        |dk  |f||¬«      S )Nr”   c                 ó–   — | t        j                  d«      z   t        j                  | «      z   d| z
  t        j                  | «      z  z   S r‹  )rP   rð   rw   rÆ  r[  )Úhalf_dfs    r6   rÒ  z*chi2_gen._entropy.<locals>.regular_formula‰  s>   € ØœbŸf™f Q›iÑ'¬"¯*©*°WÓ*=Ñ=Ø˜‘[¤B§F¡F¨7£OÑ3ñ4ð 5r8   c                 óö   — t        j                  d«      ddt        j                  dt         j                  z  «      z   z  z   }d| z  }|d|d|d|dz  z   z  z   z  z   z  dt        j                  | «      z  z   |z   S )NrU   r”   r   gUUUUUUå¿çUUUUUUÕ¿glÁlÁ¶¿g      @rï   )rð  r  Úhs      r6   r×  z-chi2_gen._entropy.<locals>.asymptotic_formula�  s   € ô —‘�q“	˜C ¤R§V¡V¨A¬b¯e©e©G£_Ñ!4Ñ5Ñ5ˆAØ�G‘ˆAØ�t˜a ¨¨5°1°S±5©=Ñ(9Ñ!9Ñ:Ñ:Ñ;ØœŸ™˜w›Ñ'ñ(Ø*+ñ,ð -r8   é}   rê  rì  )rE   r±  rð  rÒ  r×  s        r6   rò   zchi2_gen._entropy†  s4   € Ø˜‘(ˆò	5ò		-ô ˜' C™-¨'¨Ø)Ø/ô1ð 	1r8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   ry   r�   r~   r   rò   r‡   r8   r6   rÛ  rÛ  D  s<   „ ñ òBFó0ò+òSòò ò ò)òó1r8   rÛ  r´  c                   óF   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zy)Ú
cosine_gena\  A cosine continuous random variable.

    %(before_notes)s

    Notes
    -----
    The cosine distribution is an approximation to the normal distribution.
    The probability density function for `cosine` is:

    .. math::

        f(x) = \frac{1}{2\pi} (1+\cos(x))

    for :math:`-\pi \le x \le \pi`.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zcosine_gen._shape_infoµ  r¦   r8   c                 óZ   — dt         j                  z  dt        j                  |«      z   z  S ©Nr”   r   ©rP   rñ   r!  r©   s     r6   rr   zcosine_gen._pdf¸  s!   € à”R—U‘U‰{˜AœbŸf™f Q›i™KÑ(Ð(r8   c                 ór   — t        j                  |«      }t        |dk7  |fd„ t         j                   ¬«      S )Nr¿  c                 óz   — t        j                  | «      t        j                  dt         j                  z  «      z
  S r  )rP   r¦  rð   rñ   ©r  s    r6   rç  z$cosine_gen._logpdf.<locals>.<lambda>¿  s#   € ¤B§H¡H¨Q£K´"·&±&¸¼2¿5¹5¹³/Ñ$A€ r8   rÿ  )rP   r!  r   ri   r
  s      r6   rÞ   zcosine_gen._logpdf¼  s2   € Ü�F‰F�1‹IˆÜ˜!˜r™' A 4ÙAÜ%'§V¡V Gô-ð 	-r8   c                 ó,   — t        j                  |«      S rN   ©rn   Ú_cosine_cdfr©   s     r6   ru   zcosine_gen._cdfÂ  s   € Ü�‰˜qÓ!Ð!r8   c                 ó.   — t        j                  | «      S rN   rÿ  r©   s     r6   ry   zcosine_gen._sfÅ  s   € Ü�‰ ˜rÓ"Ð"r8   c                 ó,   — t        j                  |«      S rN   ©rn   Ú_cosine_invcdf©rE   rô  s     r6   r~   zcosine_gen._ppfÈ  s   € Ü×!Ñ! !Ó$Ð$r8   c                 ó.   — t        j                  |«       S rN   r  r  s     r6   r�   zcosine_gen._isfË  s   € Ü×"Ñ" 1Ó%Ð%Ð%r8   c                 óä   — t         j                  t         j                  z  dz  dz
  }dt         j                  dz  dz
  z  dt         j                  t         j                  z  dz
  dz  z  z  }d	|d	|fS )
NrD  r¶   rÃ  r$  éZ   ç      @r…  rU   rˆ   r/  )rE   rË  r  s      r6   r   zcosine_gen._statsÎ  sa   € Ü�U‰U”R—U‘U‰]˜SÑ  CÑ'ˆØ”B—E‘E˜1‘H˜r‘MÑ" c¬R¯U©U´R·U±U©]¸QÑ->ÀÑ,BÑ&BÑCˆØ�A�s˜Aˆ~Ðr8   c                 óT   — t        j                  dt         j                  z  «      dz
  S )Nr$  r‰   rï   rj   s    r6   rò   zcosine_gen._entropyÓ  s   € Ü�v‰v�aœŸ™‘g‹˜sÑ"Ð"r8   N©rƒ   r„   r…   r†   rk   rr   rÞ   ru   ry   r~   r�   r   rò   r‡   r8   r6   rö  rö     s4   „ ñò(ò)ò-ò"ò#ò%ò&òó
#r8   rö  Úcosinec                   óN   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Zd„ Zy)Ú
dgamma_genaÔ  A double gamma continuous random variable.

    The double gamma distribution is also known as the reflected gamma
    distribution [1]_.

    %(before_notes)s

    Notes
    -----
    The probability density function for `dgamma` is:

    .. math::

        f(x, a) = \frac{1}{2\Gamma(a)} |x|^{a-1} \exp(-|x|)

    for a real number :math:`x` and :math:`a > 0`. :math:`\Gamma` is the
    gamma function (`scipy.special.gamma`).

    `dgamma` takes ``a`` as a shape parameter for :math:`a`.

    %(after_notes)s

    References
    ----------
    .. [1] Johnson, Kotz, and Balakrishnan, "Continuous Univariate
           Distributions, Volume 1", Second Edition, John Wiley and Sons
           (1994).

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r	  rh   rj   s    r6   rk   zdgamma_gen._shape_infoú  r  r8   Nc                 ó�   — |j                  |¬«      }t        j                  |||¬«      }|t        j                  |dk\  dd«      z  S ©N©r×   r×  r”   r   r¿  )ÚuniformrØ  rÙ  rP   rO  )rE   r‹   r×   rØ   ÚuÚgms         r6   rÙ   zdgamma_gen._rvsý  sE   € Ø× Ñ  dÐ Ó+ˆÜ�Y‰Y�q˜t°,ˆYÓ?ˆØ”B—H‘H˜Q #™X q¨"Ó-Ñ-Ð-r8   c                 óŽ   — t        |«      }ddt        j                  |«      z  z  ||dz
  z  z  t        j                  | «      z  S r  )rŽ  rw   rØ  rP   r·   ©rE   rq   r‹   Úaxs       r6   rr   zdgamma_gen._pdf  s>   € ä�‹VˆØ�A”b—h‘h˜q“k‘MÑ" 2¨¨#©¡;Ñ.´·±¸¸³Ñ<Ð<r8   c                 ó¨   — t        |«      }t        j                  |dz
  |«      |z
  t        j                  d«      z
  t        j
                  |«      z
  S r  )rŽ  rw   rx  rP   rð   rÆ  r  s       r6   rÞ   zdgamma_gen._logpdf  s?   € Ü�‹VˆÜ�x‰x˜˜C™ Ó$ rÑ)¬B¯F©F°1«IÑ5¼¿
¹
À1»ÑEÐEr8   c           	      óš   — t        j                  |dkD  ddt        j                  ||«      z  z   dt        j                  || «      z  «      S ©Nr   r”   )rP   rO  rw   r½  rÀ  r  s      r6   ru   zdgamma_gen._cdf  sF   € Ü�x‰x˜˜A™Ø˜c¤"§+¡+¨a°Ó"3Ñ3Ñ3ØœBŸL™L¨¨Q¨BÓ/Ñ/ó1ð 	1r8   c           
      óš   — t        j                  |dkD  dt        j                  ||«      z  ddt        j                  || «      z  z   «      S r  )rP   rO  rw   rÀ  r½  r  s      r6   ry   zdgamma_gen._sf  sF   € Ü�x‰x˜˜A™ØœBŸL™L¨¨AÓ.Ñ.Ø˜c¤"§+¡+¨a°!°Ó"4Ñ4Ñ4ó6ð 	6r8   c                 ól   — t         j                  j                  |«      t        j                  d«      z
  S ©Nr”   )rò  rØ  rò   rP   rð   r  s     r6   rò   zdgamma_gen._entropy  s$   € Ü�{‰{×#Ñ# AÓ&¬¯©°«Ñ4Ð4r8   c           	      óš   — t        j                  |dkD  t        j                  |d|z  dz
  «      t        j                  |d|z  «       «      S rî   ©rP   rO  rw   rÄ  rÈ  r  s      r6   r~   zdgamma_gen._ppf  sD   € Ü�x‰x˜˜C™ÜŸ™ q¨!¨A©#°©'Ó2ÜŸ™¨¨A¨a©CÓ0Ð0ó2ð 	2r8   c           	      óš   — t        j                  |dkD  t        j                  |d|z  dz
  «       t        j                  |d|z  «      «      S rî   r   r  s      r6   r�   zdgamma_gen._isf  sD   € Ü�x‰x˜˜C™ÜŸ™¨¨1¨Q©3°©7Ó3Ð3ÜŸ™¨¨1¨Q©3Ó/ó1ð 	1r8   c                 ó<   — ||dz   z  }d|d|dz   |dz   z  |z  dz
  fS )Nr‰   rˆ   r¶   rD  r‡   )rE   r‹   rD  s      r6   r   zdgamma_gen._stats"  s4   € Ø��3‘‰iˆØ�C˜˜q ™u q¨¡u™o¨cÑ1°#Ñ5Ð5Ð5r8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   ry   rò   r~   r�   r   r‡   r8   r6   r  r  Ú  s;   „ ñò>Eó.ò
=ò
Fò1ò
6ò
5ò2ò
1ó
6r8   r  Údgammac                   óê   — e Zd ZdZej
                  Zej                  Zej                  Z
ej                  Zej                  Zej                  Zd„ Zd„ Zd„ Zd„ Zdd„Zd„ Zd	„ Zd
„ Z	d„ Zd„ Zd„ Zd„ Zy)Údpareto_lognorm_gena…  A double Pareto lognormal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `dpareto_lognorm` is:

    .. math::

        f(x, \mu, \sigma, \alpha, \beta) =
        \frac{\alpha \beta}{(\alpha + \beta) x}
        \phi\left( \frac{\log x - \mu}{\sigma} \right)
        \left( R(y_1) + R(y_2) \right)

    where :math:`R(t) = \frac{1 - \Phi(t)}{\phi(t)}`,
    :math:`\phi` and :math:`\Phi` are the normal PDF and CDF, respectively,
    :math:`y_1 = \alpha \sigma - \frac{\log x - \mu}{\sigma}`,
    and :math:`y_2 = \beta \sigma + \frac{\log x - \mu}{\sigma}`
    for real numbers :math:`x` and :math:`\mu`, :math:`\sigma > 0`,
    :math:`\alpha > 0`, and :math:`\beta > 0` [1]_.

    `dpareto_lognorm` takes
    ``u`` as a shape parameter for :math:`\mu`,
    ``s`` as a shape parameter for :math:`\sigma`,
    ``a`` as a shape parameter for :math:`\alpha`, and
    ``b`` as a shape parameter for :math:`\beta`.

    A random variable :math:`X` distributed according to the PDF above
    can be represented as :math:`X = U \frac{V_1}{V_2}` where :math:`U`,
    :math:`V_1`, and :math:`V_2` are independent, :math:`U` is lognormally
    distributed such that :math:`\log U \sim N(\mu, \sigma^2)`, and
    :math:`V_1` and :math:`V_2` follow Pareto distributions with parameters
    :math:`\alpha` and :math:`\beta`, respectively [2]_.

    %(after_notes)s

    References
    ----------
    .. [1] Hajargasht, Gholamreza, and William E. Griffiths. "Pareto-lognormal
           distributions: Inequality, poverty, and estimation from grouped income
           data." Economic Modelling 33 (2013): 593-604.
    .. [2] Reed, William J., and Murray Jorgensen. "The double Pareto-lognormal
           distribution - a new parametric model for size distributions."
           Communications in Statistics - Theory and Methods 33.8 (2004): 1733-1753.

    %(example)s

    c                 óH   — | j                  |«      | j                  |«      z  S rN   )Ú_PhicÚ_phi©rE   Úzs     r6   Ú_Rzdpareto_lognorm_gen._Rc  s   € Ø�z‰z˜!‹}˜tŸy™y¨›|Ñ+Ð+r8   c                 óH   — | j                  |«      | j                  |«      z
  S rN   )Ú_logPhicÚ_logphir)  s     r6   Ú_logRzdpareto_lognorm_gen._logRf  s   € Ø�}‰}˜QÓ $§,¡,¨q£/Ñ1Ð1r8   c           	      ó  — t        ddt        j                   t        j                  fd«      t        dddt        j                  fd«      t        dddt        j                  fd«      t        dddt        j                  fd«      gS )Nr  Fr
  r  r   r‹   rŒ   rh   rj   s    r6   rk   zdpareto_lognorm_gen._shape_infoi  sm   € Ü˜3 ¬¯©¨´·±Ð'8¸.ÓIÜ˜3 ¨¬2¯6©6 {°NÓCÜ˜3 ¨¬2¯6©6 {°NÓCÜ˜3 ¨¬2¯6©6 {°NÓCðEð 	Er8   c                 ó$   — |dkD  |dkD  z  |dkD  z  S ©Nr   r‡   )rE   r  r  r‹   rŒ   s        r6   rc   zdpareto_lognorm_gen._argchecko  s   € Ø�A‘˜!˜a™%Ñ  A¨¡EÑ*Ð*r8   Nc                 ó´   — |j                  |||¬«      }|j                  |¬«      }|j                  |¬«      }	t        j                  |||z  z   |	|z  z
  «      S ©Nr  )ÚnormalÚstandard_exponentialrP   r·   )
rE   r  r  r‹   rŒ   r×   rØ   ÚZÚE1ÚE2s
             r6   rÙ   zdpareto_lognorm_gen._rvsr  sa   € ð ×Ñ  1¨4ÐÓ0ˆØ×.Ñ.°DÐ.Ó9ˆØ×.Ñ.°DÐ.Ó9ˆÜ�v‰v�a˜"˜q™&‘j 2¨¡6Ñ)Ó*Ð*r8   c                 óf  — t        j                  dd¬«      5  t        j                  |«      |}}||z
  |z  }||z  |z
  }	||z  |z   }
t        j                  t        j                  |«      t        j                  |«      z   t        j                  ||z   «      z
  |z
  «      }|| j	                  |«      z  }|t        j
                  | j                  |	«      | j                  |
«      «      z  }d d d «       t         j                   |dk(  t        j                  |«      z  <   |d   S # 1 sw Y   Œ;xY w)Nr9  ©Úinvalidr;  r   r‡   )	rP   r<  rð   rû   r.  Ú	logaddexpr/  ri   rV   )rE   rq   r  r  r‹   rŒ   Úlog_yÚmr*  Úx1Úx2rö  s               r6   rÞ   zdpareto_lognorm_gen._logpdf{  sý   € Ü�[‰[ °(Ô;ñ 	@Ü—v‘v˜a“y !�1ˆEØ˜‘˜a‘ˆAØ�Q‘˜‘ˆBØ�Q‘˜‘ˆBÜ—*‘*œRŸV™V A›Y¬¯©°«Ñ2´R·V±V¸AÀ¹E³]ÑBÀUÑJÓKˆCØ�4—<‘< “?Ñ"ˆCØ”2—<‘< §
¡
¨2£°·
±
¸2³Ó?Ñ?ˆC÷	@ô (*§v¡v gˆˆQ�!‰V”r—x‘x “{Ñ"Ñ#Ø�2‰wˆ÷	@ð 	@ús   ˜CD'Ä'D0c           	      óò  — t        j                  dd¬«      5  t        j                  |«      |}}||z
  |z  }||z  |z
  }	||z  |z   }
| j                  |«      }| j	                  |«      }t        j                  |«      | j                  |	«      z   }t        j                  |«      | j                  |
«      z   }t        j                  ||||d«      \  }}}}}t        j                  ||g|| gdd¬«      \  }}|||z   t        j                  ||z   «      z
  g}t        j                  t        j                  ||| |z  gd¬«      «      }d d d «       t         j                   |dk(  <   |d   S # 1 sw Y   Œ%xY w)	Nr9  r;  r   r   T)rŒ   rû  Úreturn_sign)rŒ   rû  r‡   )rP   r<  rð   Ú_logPhir.  r/  rñ  rw   Ú	logsumexprû   ri   )rE   rq   r  r  r‹   rŒ   r>  r?  r*  r@  rA  rº  r»  r¼  Út4ÚoneÚt5rQ   Útemprö  s                       r6   rã   zdpareto_lognorm_gen._logcdf‡  sZ  € Ü�[‰[ °(Ô;ñ 	MÜ—v‘v˜a“y !�1ˆEØ˜‘˜a‘ˆAØ�Q‘˜‘ˆBØ�Q‘˜‘ˆBØ—‘˜a“ˆBØ—‘˜a“ˆBÜ—&‘&˜“)˜dŸj™j¨›nÑ,ˆBÜ—&‘&˜“)˜dŸj™j¨›nÑ,ˆBÜ"$×"5Ñ"5°b¸"¸bÀ"ÀaÓ"HÑˆB��B˜˜Cô Ÿ™ b¨" X°#¸°t°À1ÐRVÔW‰HˆB�Ø˜˜R™¤"§&¡&¨¨Q©£-Ñ/Ð0ˆDÜ—*‘*œRŸ\™\¨$°3¸¸¸T¹	Ð2BÈÔKÓLˆC÷	Mô  —v‘v�gˆˆA�‰F‰Ø�2‰wˆ÷#	Mð 	Mús   ˜D1E-Å-E6c           	      ó>   — t        | j                  |||||«      «      S rN   )r   rã   ©rE   rq   r  r  r‹   rŒ   s         r6   rç   zdpareto_lognorm_gen._logsf›  s   € Ü˜Ÿ™ a¨¨A¨q°!Ó4Ó5Ð5r8   c           	      óR   — t        j                  | j                  |||||«      «      S rN   rÝ  rK  s         r6   rr   zdpareto_lognorm_gen._pdf   ó"   € Ü�v‰v�d—l‘l 1 a¨¨A¨qÓ1Ó2Ð2r8   c           	      óR   — t        j                  | j                  |||||«      «      S rN   ©rP   r·   rã   rK  s         r6   ru   zdpareto_lognorm_gen._cdf£  rM  r8   c           	      óR   — t        j                  | j                  |||||«      «      S rN   r6  rK  s         r6   ry   zdpareto_lognorm_gen._sf¦  s"   € Ü�v‰v�d—k‘k ! Q¨¨1¨aÓ0Ó1Ð1r8   c                 óè   — |t        |«      }}||z  ||z
  ||z   z  z  t        j                  ||z  |dz  |dz  z  dz  z   «      z  }t        j                  |«      }t        j                  |||k  <   |S r  )ÚfloatrP   r·   rû   r  )	rE   rb   r  r  r‹   rŒ   r?  r  rö  s	            r6   r  zdpareto_lognorm_gen._munp©  su   € Ø”%˜“(ˆ1ˆØ�1‰u˜!˜a™% A¨¡EÑ*Ñ+¬b¯f©f°Q¸±U¸QÀ!¹VÀaÈ1Áf¹_ÈqÑ=PÑ5PÓ.QÑQˆÜ�j‰j˜‹oˆÜ—f‘fˆˆA�‰F‰Øˆ
r8   r  )rƒ   r„   r…   r†   r  rÞ   r.  rã   rD  rç   r-  rr   r(  ru   Ú_Phiry   r'  r+  r/  rk   rc   rÙ   r  r‡   r8   r6   r%  r%  *  s}   „ ñ0ðb �l‰l€GØ�l‰l€GØ�{‰{€HØ�9‰9€DØ�9‰9€DØ�H‰H€Eò,ò2òEò+ó+ò
òò(6ò
3ò3ò2ór8   r%  Údpareto_lognormc                   óT   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Zd„ Zd„ Zy)Údweibull_genav  A double Weibull continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `dweibull` is given by

    .. math::

        f(x, c) = c / 2 |x|^{c-1} \exp(-|x|^c)

    for a real number :math:`x` and :math:`c > 0`.

    `dweibull` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zdweibull_gen._shape_infoÊ  r  r8   Nc                 ó�   — |j                  |¬«      }t        j                  |||¬«      }|t        j                  |dk\  dd«      z  S r  )r  Úweibull_minrÙ  rP   rO  )rE   r  r×   rØ   r  Úws         r6   rÙ   zdweibull_gen._rvsÍ  sE   € Ø× Ñ  dÐ Ó+ˆÜ�O‰O˜A D°|ˆOÓDˆØ”B—H‘H˜Q #™X q¨"Ó-Ñ.Ð.r8   c                 ól   — t        |«      }|dz  ||dz
  z  z  t        j                  ||z   «      z  }|S ©Nr¶   r‰   )rŽ  rP   r·   )rE   rq   r  r  ÚPxs        r6   rr   zdweibull_gen._pdfÒ  s9   € ä�‹VˆØ�‰W�r˜A˜c™E‘{Ñ"¤R§V¡V¨R°©U¨F£^Ñ3ˆØˆ	r8   c                 ó®   — t        |«      }t        j                  |«      t        j                  d«      z
  t        j                  |dz
  |«      z   ||z  z
  S r\  )rŽ  rP   rð   rw   rx  )rE   rq   r  r  s       r6   rÞ   zdweibull_gen._logpdfØ  sC   € Ü�‹VˆÜ�v‰v�a‹yœ2Ÿ6™6 #›;Ñ&¬¯©°!°c±'¸2Ó)>Ñ>ÀÀQÁÑFÐFr8   c                 ó†   — dt        j                  t        |«      |z   «      z  }t        j                  |dkD  d|z
  |«      S ©Nr”   r   r   )rP   r·   rŽ  rO  )rE   rq   r  ÚCx1s       r6   ru   zdweibull_gen._cdfÜ  s:   € Ø”B—F‘FœC ›F A™I˜:Ó&Ñ&ˆÜ�x‰x˜˜A™˜q 3™w¨Ó,Ð,r8   c                 óÒ   — dt        j                  |dk  |d|z
  «      z  }t        j                  t        j                  |«       d|z  «      }t        j                  |dkD  || «      S ©Nr¶   r”   r‰   )rP   rO  rÌ  rð   )rE   r}   r  r«  s       r6   r~   zdweibull_gen._ppfà  sX   € Ø”2—8‘8˜A ™H a¨¨a©Ó0Ñ0ˆÜ�h‰hœŸ™˜s›�| S¨1¡WÓ-ˆÜ�x‰x˜˜C™  s dÓ+Ð+r8   c                 ó¨   — dt         j                  j                  t        j                  |«      |«      z  }t        j
                  |dkD  |d|z
  «      S r`  )rò  rY  ry   rP   rŽ  rO  )rE   rq   r  Úhalf_weibull_min_sfs       r6   ry   zdweibull_gen._sfå  sF   € Ø!¤E×$5Ñ$5×$9Ñ$9¼"¿&¹&À»)ÀQÓ$GÑGÐÜ�x‰x˜˜A™Ð2°AÐ8KÑ4KÓLÐLr8   c                 ó¸   — dt        j                  |dk  |d|z
  «      z  }t        j                  j	                  ||«      }t        j                  |dkD  | |«      S rc  )rP   rO  rò  rY  r�   )rE   r}   r  Údouble_qÚweibull_min_isfs        r6   r�   zdweibull_gen._isfé  sS   € ØœŸ™  c¡¨1¨b°1©fÓ5Ñ5ˆÜ×+Ñ+×0Ñ0°¸1Ó=ˆÜ�x‰x˜˜C™ /Ð!1°?ÓCÐCr8   c                 óP   — d|dz  z
  t        j                  dd|z  |z  z   «      z  S )Nr   rU   r‰   ©rw   rØ  r|  s      r6   r  zdweibull_gen._munpî  s+   € Ø�Q˜‘U‘œrŸx™x¨¨c°A©g¸©kÑ(9Ó:Ñ:Ð:r8   c                  ó   — y©N)r   Nr   Nr‡   r~  s     r6   r   zdweibull_gen._statsô  ó   € Ør8   c                 óp   — t         j                  j                  |«      t        j                  d«      z
  }|S r  )rò  rY  rò   rP   rð   )rE   r  ró  s      r6   rò   zdweibull_gen._entropy÷  s*   € Ü×Ñ×&Ñ& qÓ)¬B¯F©F°3«KÑ7ˆØˆr8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   r~   ry   r�   r  r   rò   r‡   r8   r6   rV  rV  ´  sB   „ ñò*Eó/ò
òGò-ò,ò
MòDò
;ò ór8   rV  Údweibullc                   ó~   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Zd„ Zd„ Ze eed¬«      d„ «       «       Zy)Ú	expon_genaE  An exponential continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `expon` is:

    .. math::

        f(x) = \exp(-x)

    for :math:`x \ge 0`.

    %(after_notes)s

    A common parameterization for `expon` is in terms of the rate parameter
    ``lambda``, such that ``pdf = lambda * exp(-lambda * x)``. This
    parameterization corresponds to using ``scale = 1 / lambda``.

    The exponential distribution is a special case of the gamma
    distributions, with gamma shape parameter ``a = 1``.

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zexpon_gen._shape_info  r¦   r8   Nc                 ó$   — |j                  |«      S rN   )r6  rÖ   s      r6   rÙ   zexpon_gen._rvs  s   € Ø×0Ñ0°Ó6Ð6r8   c                 ó.   — t        j                  | «      S rN   ©rP   r·   r©   s     r6   rr   zexpon_gen._pdf   s   € ä�v‰v�q�b‹zÐr8   c                 ó   — | S rN   r‡   r©   s     r6   rÞ   zexpon_gen._logpdf$  ó	   € Øˆrˆ	r8   c                 ó0   — t        j                  | «       S rN   ©rw   r  r©   s     r6   ru   zexpon_gen._cdf'  ó   € Ü—‘˜!˜“ˆ}Ðr8   c                 ó0   — t        j                  | «       S rN   r	  r°   s     r6   r~   zexpon_gen._ppf*  rz  r8   c                 ó.   — t        j                  | «      S rN   ru  r©   s     r6   ry   zexpon_gen._sf-  s   € Ü�v‰v�q�b‹zÐr8   c                 ó   — | S rN   r‡   r©   s     r6   rç   zexpon_gen._logsf0  rw  r8   c                 ó.   — t        j                  |«       S rN   r2  r°   s     r6   r�   zexpon_gen._isf3  ó   € Ü—‘�q“	ˆzÐr8   c                  ó   — y)N)r‰   r‰   r¶   ç      @r‡   rj   s    r6   r   zexpon_gen._stats6  rì   r8   c                  ó   — yr  r‡   rj   s    r6   rò   zexpon_gen._entropy9  ó   € Ør8   zú        When `method='MLE'`,
        this function uses explicit formulas for the maximum likelihood
        estimation of the exponential distribution parameters, so the
        `optimizer`, `loc` and `scale` keyword arguments are
        ignored.

ró   c                 ó  — t        |«      dkD  rt        d«      ‚|j                  dd «      }|j                  dd «      }t        |«       |�|�t	        d«      ‚t        j                  |«      }t        j                  |«      j                  «       st	        d«      ‚|j                  «       }|€|}n#|}||k  rt        d|t
        j                  ¬«      ‚|€|j                  «       |z
  }n|}t        |«      t        |«      fS )	Nr   úToo many arguments.rö   r÷   rø   rù   ÚexponrŸ  )r¤  r4   r3   r7   rú   rP   rû   rü   rý   ÚminrK  ri   rþ   rR  )	rE   rF   rG   r5   rö   r÷   Údata_minr.   r/   s	            r6   rC   zexpon_gen.fit<  sô   € ô ˆt‹9�qŠ=ÜÐ1Ó2Ð2à�x‰x˜ Ó%ˆØ—‘˜( DÓ)ˆä$ TÔ*àÐ Ð 2äð )ó *ð *ô �z‰z˜$Óˆä�{‰{˜4Ó ×$Ñ$Ô&ÜÐCÓDÐDà—8‘8“:ˆàˆ<à‰CàˆCØ˜#Š~ä" 7°$¼b¿f¹fÔEÐEàˆ>à—I‘I“K #Ñ%‰EàˆEô �S‹zœ5 ›<Ð'Ð'r8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   r~   ry   rç   r�   r   rò   rK   r
   r   rC   r‡   r8   r6   rq  rq  ÿ  sf   „ ñò4ó7òòòòòòòò"òð Ù ð 6ô ñ&(óó ñ&(r8   rq  r†  c                   ó<   — e Zd ZdZd„ Zd
d„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
y)Úexponnorm_genaè  An exponentially modified Normal continuous random variable.

    Also known as the exponentially modified Gaussian distribution [1]_.

    %(before_notes)s

    Notes
    -----
    The probability density function for `exponnorm` is:

    .. math::

        f(x, K) = \frac{1}{2K} \exp\left(\frac{1}{2 K^2} - x / K \right)
                  \text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right)

    where :math:`x` is a real number and :math:`K > 0`.

    It can be thought of as the sum of a standard normal random variable
    and an independent exponentially distributed random variable with rate
    ``1/K``.

    %(after_notes)s

    An alternative parameterization of this distribution (for example, in
    the Wikipedia article [1]_) involves three parameters, :math:`\mu`,
    :math:`\lambda` and :math:`\sigma`.

    In the present parameterization this corresponds to having ``loc`` and
    ``scale`` equal to :math:`\mu` and :math:`\sigma`, respectively, and
    shape parameter :math:`K = 1/(\sigma\lambda)`.

    .. versionadded:: 0.16.0

    References
    ----------
    .. [1] Exponentially modified Gaussian distribution, Wikipedia,
           https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS )NÚKFr   r
  rh   rj   s    r6   rk   zexponnorm_gen._shape_info™  r  r8   Nc                 óV   — |j                  |«      |z  }|j                  |«      }||z   S rN   )r6  rÕ   )rE   rŒ  r×   rØ   ÚexpvalÚgvals         r6   rÙ   zexponnorm_gen._rvsœ  s1   € Ø×2Ñ2°4Ó8¸1Ñ<ˆØ×+Ñ+¨DÓ1ˆØ˜‰}Ðr8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  )rE   rq   rŒ  s      r6   rr   zexponnorm_gen._pdf¡  ó   € Ü�v‰v�d—l‘l 1 aÓ(Ó)Ð)r8   c                 óp   — d|z  }|d|z  |z
  z  }|t        ||z
  «      z   t        j                  |«      z
  S ©Nr‰   r”   ©rÃ   rP   rð   )rE   rq   rŒ  ÚinvKÚexpargs        r6   rÞ   zexponnorm_gen._logpdf¤  s>   € Ø�Q‰wˆØ˜˜t™ a™Ñ(ˆØœ Q¨¡XÓ.Ñ.´·±¸³Ñ:Ð:r8   c                 ó†   — d|z  }|d|z  |z
  z  }|t        ||z
  «      z   }t        |«      t        j                  |«      z
  S r“  ©rÃ   rÀ   rP   r·   ©rE   rq   rŒ  r•  rŽ  Úlogprods         r6   ru   zexponnorm_gen._cdf©  sG   € Ø�Q‰wˆØ˜˜t™ a™Ñ(ˆØœ<¨¨D©Ó1Ñ1ˆÜ˜‹|œbŸf™f W›oÑ-Ð-r8   c                 óˆ   — d|z  }|d|z  |z
  z  }|t        ||z
  «      z   }t        | «      t        j                  |«      z   S r“  r˜  r™  s         r6   ry   zexponnorm_gen._sf¯  sI   € Ø�Q‰wˆØ˜˜t™ a™Ñ(ˆØœ<¨¨D©Ó1Ñ1ˆÜ˜!˜‹}œrŸv™v g›Ñ.Ð.r8   c                 óZ   — ||z  }d|z   }d|dz  z  |dz  z  }d|z  |z  |dz  z  }||||fS )Nr‰   rU   r†  rA  r�  r  r‡   )rE   rŒ  ÚK2ÚopK2ÚskwÚkrts         r6   r   zexponnorm_gen._statsµ  sO   € Ø�‰UˆØ�R‰xˆØ�!�Q‘$‰h˜ ™Ñ%ˆØ�B‰h˜‰m˜d R™jÑ(ˆØ�$˜˜SÐ Ð r8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   ry   r   r‡   r8   r6   rŠ  rŠ  o  s,   „ ñ(òREóò
*ò;ò
.ò/ó!r8   rŠ  Ú	exponnormc                 óT   — t        j                  t        j                  || «      «      S )a'  
    Compute (1 + x)**y - 1.

    Uses expm1 and xlog1py to avoid loss of precision when
    (1 + x)**y is close to 1.

    Note that the inverse of this function with respect to x is
    ``_pow1pm1(x, 1/y)``.  That is, if

        t = _pow1pm1(x, y)

    then

        x = _pow1pm1(t, 1/y)
    )rP   r  rw   rw  ©rq   r�  s     r6   Ú_pow1pm1r¤  À  s   € ô  �8‰8”B—J‘J˜q !Ó$Ó%Ð%r8   c                   ó:   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
y	)
Úexponweib_gena   An exponentiated Weibull continuous random variable.

    %(before_notes)s

    See Also
    --------
    weibull_min, numpy.random.Generator.weibull

    Notes
    -----
    The probability density function for `exponweib` is:

    .. math::

        f(x, a, c) = a c [1-\exp(-x^c)]^{a-1} \exp(-x^c) x^{c-1}

    and its cumulative distribution function is:

    .. math::

        F(x, a, c) = [1-\exp(-x^c)]^a

    for :math:`x > 0`, :math:`a > 0`, :math:`c > 0`.

    `exponweib` takes :math:`a` and :math:`c` as shape parameters:

    * :math:`a` is the exponentiation parameter,
      with the special case :math:`a=1` corresponding to the
      (non-exponentiated) Weibull distribution `weibull_min`.
    * :math:`c` is the shape parameter of the non-exponentiated Weibull law.

    %(after_notes)s

    References
    ----------
    https://en.wikipedia.org/wiki/Exponentiated_Weibull_distribution

    %(example)s

    c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS ©Nr‹   Fr   r
  r  rh   ©rE   ri  r!  s      r6   rk   zexponweib_gen._shape_infoü  rk  r8   c                 óN   — t        j                  | j                  |||«      «      S rN   rÝ  ©rE   rq   r‹   r  s       r6   rr   zexponweib_gen._pdf	  ó    € ô �v‰v�d—l‘l 1 a¨Ó+Ó,Ð,r8   c                 ó  — ||z   }t        j                  |«       }t        j                  |«      t        j                  |«      z   t        j                  |dz
  |«      z   |z   t        j                  |dz
  |«      z   }|S r  )rw   r  rP   rð   rx  )rE   rq   r‹   r  ÚnegxcÚexm1cÚlogps          r6   rÞ   zexponweib_gen._logpdf	  so   € Ø�A‘�ˆÜ—‘˜%“Ð ˆÜ—‘�q“	œBŸF™F 1›IÑ%¬¯©°°S±¸%Ó(@Ñ@ØñÜŸ™  S¡¨!Ó,ñ-ˆàˆr8   c                 ó@   — t        j                  ||z   «       }||z  S rN   ry  )rE   rq   r‹   r  r¯  s        r6   ru   zexponweib_gen._cdf	  s!   € Ü—‘˜1˜a™4˜%“Ð ˆØ�a‰xˆr8   c                 ón   — t        j                  |d|z  z   «       t        j                  d|z  «      z  S r  )rw   r¦  rP   rû   )rE   r}   r‹   r  s       r6   r~   zexponweib_gen._ppf	  s0   € Ü—‘˜1˜s 1™u™:˜+Ó&Ð&¬¯©°C¸±EÓ):Ñ:Ð:r8   c                 óL   — t        t        j                  ||z   «       |«       S rN   )r¤  rP   r·   r«  s       r6   ry   zexponweib_gen._sf	  s"   € Üœ"Ÿ&™& ! Q¡$ ›-˜¨Ó+Ð+Ð+r8   c                 óX   — t        j                  t        | d|z  «       «       d|z  z  S r^   )rP   rð   r¤  )rE   rô  r‹   r  s       r6   r�   zexponweib_gen._isf	  s-   € Ü—‘œ 1 " a¨¡cÓ*Ð*Ó+Ð+¨q°©sÑ3Ð3r8   N©rƒ   r„   r…   r†   rk   rr   rÞ   ru   r~   ry   r�   r‡   r8   r6   r¦  r¦  Ó  s+   „ ñ'òPò
-ò
òò;ò,ó4r8   r¦  Ú	exponweibc                   ó:   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
y	)
Úexponpow_genaƒ  An exponential power continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `exponpow` is:

    .. math::

        f(x, b) = b x^{b-1} \exp(1 + x^b - \exp(x^b))

    for :math:`x \ge 0`, :math:`b > 0`.  Note that this is a different
    distribution from the exponential power distribution that is also known
    under the names "generalized normal" or "generalized Gaussian".

    `exponpow` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    References
    ----------
    http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Exponentialpower.pdf

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS ©NrŒ   Fr   r
  rh   rj   s    r6   rk   zexponpow_gen._shape_info:	  r  r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  ©rE   rq   rŒ   s      r6   rr   zexponpow_gen._pdf=	  ó   € ä�v‰v�d—l‘l 1 aÓ(Ó)Ð)r8   c                 ó¦   — ||z  }dt        j                  |«      z   t        j                  |dz
  |«      z   |z   t        j                  |«      z
  }|S ©Nr   r‰   )rP   rð   rw   rx  r·   )rE   rq   rŒ   Úxbr�  s        r6   rÞ   zexponpow_gen._logpdfA	  sG   € Ø�‰TˆØ”—‘�q“	‰MœBŸH™H Q¨¡W¨aÓ0Ñ0°2Ñ5¼¿¹¸r»
ÑBˆØˆr8   c                 ó\   — t        j                  t        j                  ||z  «       «       S rN   ry  r¼  s      r6   ru   zexponpow_gen._cdfF	  s"   € Ü—‘œ"Ÿ(™( 1 a¡4›.˜Ó)Ð)Ð)r8   c                 óZ   — t        j                  t        j                  ||z  «       «      S rN   ©rP   r·   rw   r  r¼  s      r6   ry   zexponpow_gen._sfI	  s   € Ü�v‰v”r—x‘x  1¡“~�oÓ&Ð&r8   c                 ó`   — t        j                  t        j                  |«       «      d|z  z  S r  ©rw   r¦  rP   rð   r¼  s      r6   r�   zexponpow_gen._isfL	  s$   € Ü—‘œ"Ÿ&™& ›)˜Ó$¨¨1©Ñ-Ð-r8   c                 óp   — t        t        j                  t        j                  | «       «      d|z  «      S r  ©Úpowrw   r¦  ©rE   r}   rŒ   s      r6   r~   zexponpow_gen._ppfO	  s(   € Ü”2—8‘8œRŸX™X q b›\˜MÓ*¨C°©EÓ2Ð2r8   N)rƒ   r„   r…   r†   rk   rr   rÞ   ru   ry   r�   r~   r‡   r8   r6   r¸  r¸  	  s+   „ ñò6Eò*òò
*ò'ò.ó3r8   r¸  Úexponpowc                   ó`   — e Zd ZdZej
                  Zd„ Zdd„Zd„ Z	d„ Z
d„ Zd„ Zd	„ Zd
„ Zd„ Zy)Úfatiguelife_gena0  A fatigue-life (Birnbaum-Saunders) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `fatiguelife` is:

    .. math::

        f(x, c) = \frac{x+1}{2c\sqrt{2\pi x^3}} \exp(-\frac{(x-1)^2}{2x c^2})

    for :math:`x >= 0` and :math:`c > 0`.

    `fatiguelife` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    .. [1] "Birnbaum-Saunders distribution",
           https://en.wikipedia.org/wiki/Birnbaum-Saunders_distribution

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zfatiguelife_gen._shape_infos	  r  r8   Nc                 ó�   — |j                  |«      }d|z  |z  }||z  }dd|z  z   d|z  t        j                  d|z   «      z  z   }|S )Nr”   r‰   rU   r   )rÕ   rP   rÿ   )rE   r  r×   rØ   r*  rq   rA  Úts           r6   rÙ   zfatiguelife_gen._rvsv	  sT   € Ø×(Ñ(¨Ó.ˆØ�‰E�!‰GˆØˆq‰SˆØ�!�B‘$‰J˜˜1™œRŸW™W Q¨¡V›_Ñ,Ñ,ˆØˆr8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  r
  s      r6   rr   zfatiguelife_gen._pdf}	  s   € ô �v‰v�d—l‘l 1 aÓ(Ó)Ð)r8   c                 ó  — t        j                  |dz   «      |dz
  dz  d|z  |dz  z  z  z
  t        j                  d|z  «      z
  dt        j                  dt         j                  z  «      dt        j                  |«      z  z   z  z
  S )Nr   rU   r¶   r”   r†  rï   r
  s      r6   rÞ   zfatiguelife_gen._logpdf‚	  ss   € Ü—‘�q˜‘s“˜q ™s Q™h¨#¨a©%°°1±©*Ñ5Ñ5¼¿¹¸qÀ¹s»ÑCØ”R—V‘V˜AœbŸe™e™G“_ q¬¯©°«¡{Ñ2Ñ3ñ4ð 	5r8   c                 ó|   — t        d|z  t        j                  |«      dt        j                  |«      z  z
  z  «      S r  )rÀ   rP   rÿ   r
  s      r6   ru   zfatiguelife_gen._cdf†	  s/   € Ü˜˜q™¤B§G¡G¨A£J°´R·W±W¸Q³Z±Ñ$?Ñ@ÓAÐAr8   c                 óf   — |t        |«      z  }d|t        j                  |dz  dz   «      z   dz  z  S ©Nç      Ð?rU   r$  ©rÇ   rP   rÿ   ©rE   r}   r  Útmps       r6   r~   zfatiguelife_gen._ppf‰	  s6   € Ø”)˜A“,ÑˆØ�sœRŸW™W S¨!¡V¨a¡ZÓ0Ñ0°1Ñ4Ñ4Ð4r8   c                 ó|   — t        d|z  t        j                  |«      dt        j                  |«      z  z
  z  «      S r  )rÊ   rP   rÿ   r
  s      r6   ry   zfatiguelife_gen._sf�	  s/   € Ü˜˜a™¤2§7¡7¨1£:°´B·G±G¸A³J±Ñ#>Ñ?Ó@Ð@r8   c                 óh   — | t        |«      z  }d|t        j                  |dz  dz   «      z   dz  z  S rÔ  rÖ  r×  s       r6   r�   zfatiguelife_gen._isf�	  s8   € Øˆb”9˜Q“<ÑˆØ�sœRŸW™W S¨!¡V¨a¡ZÓ0Ñ0°1Ñ4Ñ4Ð4r8   c                 óº   — ||z  }|dz  dz   }d|z  dz   }||z  dz  }d|z  d|z  dz   z  t        j                  |d«      z  }d	|z  d
|z  dz   z  |dz  z  }||||fS )Nr¶   r‰   r	  rJ  r$  é   r�  rÊ  r…  é]   g      D@©rP   rÌ  )rE   r  Úc2rC  ÚdenrD  rE  rF  s           r6   r   zfatiguelife_gen._stats”	  s‡   € ð ˆq‰SˆØ�#‰X˜‰^ˆØ�B‰h˜‰nˆØ�‰f�s‰lˆØ�‰U�b˜‘e˜c‘kÑ"¤R§X¡X¨c°3Ó%7Ñ7ˆØ�‰V�r˜"‘u˜t‘|Ñ$ s¨C¡xÑ/ˆØ�3˜˜BˆÐr8   r  )rƒ   r„   r…   r†   r   r  r  rk   rÙ   rr   rÞ   ru   r~   ry   r�   r   r‡   r8   r6   rÌ  rÌ  V	  sD   „ ñð4 "×4Ñ4€MòEóò*ò
5òBò5òAò5ór8   rÌ  Úfatiguelifec                   ó<   — e Zd ZdZd„ Zd„ Zd
d„Zd„ Zd„ Zd„ Z	d	„ Z
y)Úfoldcauchy_genao  A folded Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `foldcauchy` is:

    .. math::

        f(x, c) = \frac{1}{\pi (1+(x-c)^2)} + \frac{1}{\pi (1+(x+c)^2)}

    for :math:`x \ge 0` and :math:`c \ge 0`.

    `foldcauchy` takes ``c`` as a shape parameter for :math:`c`.

    %(example)s

    c                 ó   — |dk\  S r2  r‡   r~  s     r6   rc   zfoldcauchy_gen._argcheckº	  ó   € Ø�A‰vˆr8   c                 ó@   — t        dddt        j                  fd«      gS ©Nr  Fr   rg   rh   rj   s    r6   rk   zfoldcauchy_gen._shape_info½	  ó   € Ü˜3 ¨¬2¯6©6 {°MÓBÐCÐCr8   Nc                 óD   — t        t        j                  |||¬«      «      S )N©r.   r×   rØ   )rŽ  r¬  rÙ  ©rE   r  r×   rØ   s       r6   rÙ   zfoldcauchy_gen._rvsÀ	  s&   € Ü”6—:‘: !¨$Ø+7ð ó 9ó :ð 	:r8   c                 ód   — dt         j                  z  dd||z
  dz  z   z  dd||z   dz  z   z  z   z  S ©Nr‰   r   rU   r/  r
  s      r6   rr   zfoldcauchy_gen._pdfÄ	  s<   € à”2—5‘5‰y˜#˜q ! A¡#¨¡™zÑ*¨S°!°Q°q±S¸1±H±*Ñ-=Ñ=Ñ>Ð>r8   c                 óŒ   — dt         j                  z  t        j                  ||z
  «      t        j                  ||z   «      z   z  S r  ©rP   rñ   Úarctanr
  s      r6   ru   zfoldcauchy_gen._cdfÈ	  s2   € Ø”2—5‘5‰yœ"Ÿ)™) A a¡C›.¬2¯9©9°Q°q±S«>Ñ9Ñ:Ð:r8   c                 óŠ   — t        j                  d||z
  «      t        j                  d||z   «      z   t         j                  z  S r^   r‘  r
  s      r6   ry   zfoldcauchy_gen._sfË	  s6   € ô
 —
‘
˜1˜a !™eÓ$¤r§z¡z°!°Q¸±UÓ';Ñ;¼R¿U¹UÑBÐBr8   c                 ó~   — t         j                  t         j                  t         j                  t         j                  fS rN   r  r~  s     r6   r   zfoldcauchy_gen._statsÒ	  rœ  r8   r  ©rƒ   r„   r…   r†   rc   rk   rÙ   rr   ru   ry   r   r‡   r8   r6   rã  rã  ¦	  s,   „ ñò&òDó:ò?ò;òCó.r8   rã  Ú
foldcauchyc                   óH   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Zy)Úf_genaù  An F continuous random variable.

    For the noncentral F distribution, see `ncf`.

    %(before_notes)s

    See Also
    --------
    ncf

    Notes
    -----
    The F distribution with :math:`df_1 > 0` and :math:`df_2 > 0` degrees of freedom is
    the distribution of the ratio of two independent chi-squared distributions with
    :math:`df_1` and :math:`df_2` degrees of freedom, after rescaling by
    :math:`df_2 / df_1`.

    The probability density function for `f` is:

    .. math::

        f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}}
                                {(df_2+df_1 x)^{(df_1+df_2)/2}
                                 B(df_1/2, df_2/2)}

    for :math:`x > 0`.

    `f` accepts shape parameters ``dfn`` and ``dfd`` for :math:`df_1`, the degrees of
    freedom of the chi-squared distribution in the numerator, and :math:`df_2`, the
    degrees of freedom of the chi-squared distribution in the denominator, respectively.

    %(after_notes)s

    %(example)s

    c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS )NÚdfnFr   r
  Údfdrh   )rE   ÚidfnÚidfds      r6   rk   zf_gen._shape_infoþ	  s<   € Ü˜% ¨¬B¯F©F¨°^ÓDˆÜ˜% ¨¬B¯F©F¨°^ÓDˆØ�dˆ|Ðr8   Nc                 ó(   — |j                  |||«      S rN   )r�  )rE   rø  rù  r×   rØ   s        r6   rÙ   z
f_gen._rvs
  s   € Ø�~‰~˜c 3¨Ó-Ð-r8   c                 óN   — t        j                  | j                  |||«      «      S rN   rÝ  ©rE   rq   rø  rù  s       r6   rr   z
f_gen._pdf
  s    € ô �v‰v�d—l‘l 1 c¨3Ó/Ó0Ð0r8   c                 óF  — d|z  }d|z  }|dz  t        j                  |«      z  |dz  t        j                  |«      z  z   t        j                  |dz  dz
  |«      z   ||z   dz  t        j                  |||z  z   «      z  t        j                  |dz  |dz  «      z   z
  }|S ©Nr‰   rU   r   )rP   rð   rw   rx  ry  )rE   rq   rø  rù  rb   r?  rz  s          r6   rÞ   zf_gen._logpdf
  s™   € Ø�#‰IˆØ�#‰IˆØ�‰s”R—V‘V˜A“Y‰  1¡¤r§v¡v¨a£y¡Ñ0´2·8±8¸A¸a¹CÀ!¹GÀQÓ3GÑGØ�a‘C˜‘7œbŸf™f Q¨¨1©¡W›oÑ-´·	±	¸!¸A¹#¸qÀ¹sÓ0CÑCñEˆàˆ
r8   c                 ó0   — t        j                  |||«      S rN   )rw   Úfdtrrþ  s       r6   ru   z
f_gen._cdf
  s   € Ü�w‰w�s˜C Ó#Ð#r8   c                 ó0   — t        j                  |||«      S rN   )rw   Úfdtrcrþ  s       r6   ry   z	f_gen._sf
  ó   € Ü�x‰x˜˜S !Ó$Ð$r8   c                 ó0   — t        j                  |||«      S rN   )rw   Úfdtri)rE   r}   rø  rù  s       r6   r~   z
f_gen._ppf
  r  r8   c                 ó¤  — d|z  d|z  }}|dz
  |dz
  |dz
  |dz
  f\  }}}}t        |dkD  ||fd„ t        j                  «      }	t        |dkD  ||||fd	„ t        j                  «      }
t        |d
kD  ||||fd„ t        j                  «      }|t        j                  d«      z  }t        |dkD  |||fd„ t        j                  «      }|dz  }|	|
||fS )Nr‰   r¶   rJ  r�  ç       @rU   c                 ó   — | |z  S rN   r‡   )Úv2Úv2_2s     r6   rç  zf_gen._stats.<locals>.<lambda>"
  s
   € ˜R $™Y€ r8   r$  c                 ó6   — d|z  |z  | |z   z  | |dz  z  |z  z  S r  r‡   )Úv1r  r  Úv2_4s       r6   rç  zf_gen._stats.<locals>.<lambda>'
  s-   € Ø�‰F�R‰K˜2 ™9Ñ%¨¨d°A©g©¸Ñ)<Ñ=ð r8   r…  c                 óV   — d| z  |z   |z  t        j                  || | |z   z  z  «      z  S r  r‡  )r  r  r  Úv2_6s       r6   rç  zf_gen._stats.<locals>.<lambda>-
  s3   € Ø�‰V�d‰]˜dÑ"¤R§W¡W¨T°R¸2À¹9Ñ5EÑ-FÓ%GÑGð r8   r.  c                 ó   — d| | z  |z  z   |z  S )Nr.  r‡   )rE  r  Úv2_8s      r6   rç  zf_gen._stats.<locals>.<lambda>4
  s   €  A¨¨R©°$©Ñ$6¸$Ñ#>€ r8   rÊ  )r   rP   ri   r  rÿ   )rE   rø  rù  r  r  r  r  r  r  rC  rD  rE  rF  s                r6   r   zf_gen._stats
  s  € Ø�c‘˜2 ™8ˆBˆØ!# b¡¨"¨r©'°2¸±7¸BÀ¹GÐ!CÑˆˆd�D˜$äØ�‰F�R˜�JÙ&Ü�F‰Fóˆô
 Ø�‰F�R˜˜T 4Ð(ñ>ä�F‰Fó	ˆô Ø�‰F�R˜˜t TÐ*ñHä�F‰Fó	ˆð
 	Œb�g‰g�b‹kÑˆäØ�‰F�R˜˜tÐ$Ù>Ü�F‰Fóˆð 	ˆg‰ˆà�3˜˜BˆÐr8   c                 óL  — d|z  }d|z  }d||z   z  }t        j                  |«      t        j                  |«      z
  t        j                  ||«      z   d|z
  t        j                  |«      z  z   d|z   t        j                  |«      z  z
  |t        j                  |«      z  z   S rù  )rP   rð   rw   ry  r[  )rE   rø  rù  Úhalf_dfnÚhalf_dfdÚhalf_sums         r6   rò   zf_gen._entropy:
  sŸ   € ð ˜‘9ˆØ˜‘9ˆØ˜# ™)Ñ$ˆä—‘�s“œbŸf™f S›kÑ)¬B¯I©I°hÀÓ,IÑIØ�X‘¤§¡¨Ó!1Ñ1ñ2Ø56¸±\Ü—‘�xÓ ñ5!ñ!à#+¬b¯f©f°XÓ.>Ñ#>ñ?ð 	@r8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   ry   r~   r   rò   r‡   r8   r6   rö  rö  Ù	  s6   „ ñ#òHó
.ò1òò$ò%ò%òó<
@r8   rö  r�  c                   ó<   — e Zd ZdZd„ Zd„ Zd
d„Zd„ Zd„ Zd„ Z	d	„ Z
y)Úfoldnorm_genaz  A folded normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `foldnorm` is:

    .. math::

        f(x, c) = \sqrt{2/\pi} cosh(c x) \exp(-\frac{x^2+c^2}{2})

    for :math:`x \ge 0` and :math:`c \ge 0`.

    `foldnorm` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 ó   — |dk\  S r2  r‡   r~  s     r6   rc   zfoldnorm_gen._argcheckh
  rå  r8   c                 ó@   — t        dddt        j                  fd«      gS rç  rh   rj   s    r6   rk   zfoldnorm_gen._shape_infok
  rè  r8   Nc                 ó<   — t        |j                  |«      |z   «      S rN   ©rŽ  rÕ   rë  s       r6   rÙ   zfoldnorm_gen._rvsn
  s   € Ü�<×/Ñ/°Ó5¸Ñ9Ó:Ð:r8   c                 ó<   — t        ||z   «      t        ||z
  «      z   S rN   rÛ   r
  s      r6   rr   zfoldnorm_gen._pdfq
  s   € ä˜˜Q™Ó¤)¨A¨a©C£.Ñ0Ð0r8   c                 ó    — t        j                  d«      }dt        j                  ||z
  |z  «      t        j                  ||z   |z  «      z   z  S rÂ  )rP   rÿ   rw   Úerf)rE   rq   r  Úsqrt_twos       r6   ru   zfoldnorm_gen._cdfu
  sC   € Ü—7‘7˜1“:ˆØ”b—f‘f˜a !™e XÑ-Ó.´·±¸¸Q¹ÀÑ8HÓ1IÑIÑJÐJr8   c                 ó<   — t        ||z
  «      t        ||z   «      z   S rN   rå   r
  s      r6   ry   zfoldnorm_gen._sfy
  s   € Ü˜˜A™‹¤¨!¨a©%£Ñ0Ð0r8   c                 óà  — ||z  }t        j                  d|z  «      t        j                  dt         j                  z  «      z  }d|z  |t	        j
                  |t        j                  d«      z  «      z  z   }|dz   ||z  z
  }d||z  |z  ||z  z
  |z
  z  }|t        j                  |d«      z  }||dz   z  dz   d|z  |z  z   }|d|d	z
  z  d	|dz  z  z
  |dz  z  z  }||dz  z  d	z
  }||||fS )
Nç      à¿r¶   rU   r   rÊ  r�  r†  r	  rD  )rP   r·   rÿ   rñ   rw   r   rÌ  )rE   r  rß  ÚexpfacrC  rD  rE  rF  s           r6   r   zfoldnorm_gen._stats|
  s	  € ð ˆq‰SˆÜ—‘˜˜R™“¤2§7¡7¨2¬b¯e©e©8Ó#4Ñ4ˆà�‰Y˜œRŸV™V A¤b§g¡g¨a£j¡LÓ1Ñ1Ñ1ˆØ�1‰f�r˜"‘u‰nˆà�2�b‘5˜‘8˜b ™eÑ# fÑ,Ñ-ˆØ
Œb�h‰h�s˜CÓ Ñ ˆà�2˜‘7‰^˜aÑ " V¡)¨B¡,Ñ.ˆØ
ˆr�R˜"‘W‰~  R¨¡U¡
Ñ*¨b°!©eÑ3Ñ3ˆØ�#�s‘(‰]˜RÑˆà�3˜˜BˆÐr8   r  ró  r‡   r8   r6   r  r  R
  s,   „ ñò*òDó;ò1òKò1ór8   r  Úfoldnormc                   óx   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Z eed¬«      ˆ fd„«       Zˆ xZS )Úweibull_min_genaý  Weibull minimum continuous random variable.

    The Weibull Minimum Extreme Value distribution, from extreme value theory
    (Fisher-Gnedenko theorem), is also often simply called the Weibull
    distribution. It arises as the limiting distribution of the rescaled
    minimum of iid random variables.

    %(before_notes)s

    See Also
    --------
    weibull_max, numpy.random.Generator.weibull, exponweib

    Notes
    -----
    The probability density function for `weibull_min` is:

    .. math::

        f(x, c) = c x^{c-1} \exp(-x^c)

    for :math:`x > 0`, :math:`c > 0`.

    `weibull_min` takes ``c`` as a shape parameter for :math:`c`.
    (named :math:`k` in Wikipedia article and :math:`a` in
    ``numpy.random.weibull``).  Special shape values are :math:`c=1` and
    :math:`c=2` where Weibull distribution reduces to the `expon` and
    `rayleigh` distributions respectively.

    Suppose ``X`` is an exponentially distributed random variable with
    scale ``s``. Then ``Y = X**k`` is `weibull_min` distributed with shape
    ``c = 1/k`` and scale ``s**k``.

    %(after_notes)s

    References
    ----------
    https://en.wikipedia.org/wiki/Weibull_distribution

    https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zweibull_min_gen._shape_infoÀ
  r  r8   c                 óh   — |t        ||dz
  «      z  t        j                  t        ||«       «      z  S r^   ©rÈ  rP   r·   r
  s      r6   rr   zweibull_min_gen._pdfÃ
  s,   € à”�Q˜˜!™“‰}œRŸV™V¤S¨¨A£Y JÓ/Ñ/Ð/r8   c                 óz   — t        j                  |«      t        j                  |dz
  |«      z   t	        ||«      z
  S r^   ©rP   rð   rw   rx  rÈ  r
  s      r6   rÞ   zweibull_min_gen._logpdfÇ
  s/   € Ü�v‰v�a‹yœ2Ÿ8™8 A¨¡E¨1Ó-Ñ-´°A°q³	Ñ9Ð9r8   c                 óD   — t        j                  t        ||«       «       S rN   ©rw   r  rÈ  r
  s      r6   ru   zweibull_min_gen._cdfÊ
  s   € Ü—‘œ#˜a ›)˜Ó$Ð$Ð$r8   c                 óJ   — t        t        j                  | «       d|z  «      S r  rÇ  r  s      r6   r~   zweibull_min_gen._ppfÍ
  s   € Ü”B—H‘H˜a˜R“L�= # a¡%Ó(Ð(r8   c                 óL   — t        j                  | j                  ||«      «      S rN   r6  r
  s      r6   ry   zweibull_min_gen._sfÐ
  ó   € Ü�v‰v�d—k‘k ! QÓ'Ó(Ð(r8   c                 ó   — t        ||«       S rN   ©rÈ  r
  s      r6   rç   zweibull_min_gen._logsfÓ
  s   € Ü�A�q“	ˆzÐr8   c                 ó:   — t        j                  |«       d|z  z  S r^   r2  r  s      r6   r�   zweibull_min_gen._isfÖ
  s   € Ü—‘˜“�
˜a ™cÑ"Ð"r8   c                 ó>   — t        j                  d|dz  |z  z   «      S r  rj  r|  s      r6   r  zweibull_min_gen._munpÙ
  s   € Ü�x‰x˜˜A˜c™E !™G™Ó$Ð$r8   c                 óV   — t          |z  t        j                  |«      z
  t         z   dz   S r^   ©r$   rP   rð   r~  s     r6   rò   zweibull_min_gen._entropyÜ
  ó%   € Üˆw˜‰{œRŸV™V A›YÑ&¬Ñ/°!Ñ3Ð3r8   aÌ          If ``method='mm'``, parameters fixed by the user are respected, and the
        remaining parameters are used to match distribution and sample moments
        where possible. For example, if the user fixes the location with
        ``floc``, the parameters will only match the distribution skewness and
        variance to the sample skewness and variance; no attempt will be made
        to match the means or minimize a norm of the errors.
        

ró   c           	      ó4  •‡‡— t        |t        «      r7|j                  «       dk(  r|j                  «       }nt	        ‰| �  |g|¢­i |¤ŽS |j                  dd«      rt	        ‰| �  |g|¢­i |¤ŽS t        | |||«      \  }}}}|j                  dd«      j                  «       }d„ Št        j                  |«      Šd} ‰|«      }	‰|	k  r|dk7  r|€|st	        ‰| �  |g|¢­i |¤ŽS |dk(  rd	\  }
}}n6t        |«      r|d   nd }
|j                  d
d «      }|j                  dd «      }|€!|
€t        ˆˆfd„d|gd¬«      j                  }
n|�|}
|€h|€ft        j                   |«      }t        j"                  |t%        j&                  dd|
z  z   «      t%        j&                  dd|
z  z   «      dz  z
  z  «      }n|�|}|€9|€7t        j(                  |«      }||t%        j&                  dd|
z  z   «      z  z
  }n|�|}|dk(  r|
||fS t	        ‰| �  ||
f||dœ|¤ŽS )Nr   ÚsuperfitFr1   r;   c                 óì   — t        j                  dd| z  z   «      }t        j                  dd| z  z   «      }t        j                  dd| z  z   «      }d|dz  z  d|z  |z  z
  |z   }||dz  z
  dz  }||z  S )Nr   rU   r†  rÊ  rj  )r  Úgamma1Úgamma2Úgamma3Únumrà  s         r6   Úskewz!weibull_min_gen.fit.<locals>.skewú
  s|   € Ü—X‘X˜a  !¡™e“_ˆFÜ—X‘X˜a  !¡™e“_ˆFÜ—X‘X˜a  !¡™e“_ˆFØ�f˜a‘i‘- ! F¡(¨6¡/Ñ1°FÑ:ˆCØ˜F A™IÑ%¨Ñ-ˆCØ�s‘7ˆNr8   g     ˆÃ@r<   ©NNNr.   r/   c                 ó   •—  ‰| «      ‰z
  S rN   r‡   )r  r  rA  s    €€r6   rç  z%weibull_min_gen.fit.<locals>.<lambda>  s   ø€ ¡d¨1£g°¡k€ r8   g{®Gáz”?Úbisect)Úbracketr1   r   rU   ©r.   r/   )r?   r*   r@   r“  rA   rC   r3   Ú_check_fit_input_parametersr=   r>   rò  rA  r¤  r+   ÚrootrP   r§  rÿ   rw   rØ  rþ   )rE   rF   rG   r5   Úfcrö   r÷   r1   Úmax_cÚs_minr  r.   r/   rË  r?  r  rA  r–  s                  @@€r6   rC   zweibull_min_gen.fitß
  s<  ú€ ô �dœLÔ)Ø× Ñ Ó" aÒ'Ø—~‘~Ó'‘ä‘w‘{ 4Ð7¨$Ò7°$Ñ7Ð7à�8‰8�J Ô&Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ô "=¸TÀ4Ø=AÀ4ó"IÑˆˆb�$˜à—‘˜( EÓ*×0Ñ0Ó2ˆò	ô �J‰J�tÓˆØˆÙ�U“ˆØˆuŠ9˜ 4š¨B¨J¹tÜ‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ð �TŠ>Ø,‰MˆAˆs‘Eä˜tœ9��Q’¨$ˆAØ—(‘(˜5 $Ó'ˆCØ—H‘H˜W dÓ+ˆEàˆ:˜!˜)ô Ô1¸DÀ%¸=Ø#+ô-ß-1©Tñ àˆ^ØˆAàˆ>˜e˜mÜ—‘�t“ˆAÜ—G‘G˜A¤§¡¨!¨A¨a©C©%£´2·8±8¸A¸aÀ¹c¹E³?ÀAÑ3EÑ!EÑFÓG‰EØÐØˆEàˆ<˜C˜KÜ—‘˜“ˆAØ�eœBŸH™H Q¨¨1©¡WÓ-Ñ-Ñ-‰CØÐØˆCà�TŠ>Ø�c˜5�=Ð ô ‘7‘;˜t QÐE¨C°uÑEÀÑEÐEr8   )rƒ   r„   r…   r†   rk   rr   rÞ   ru   r~   ry   rç   r�   r  rò   r	   r   rC   rÐ  rÑ  s   @r6   r(  r(  “
  s`   ø„ ñ+òXEò0ò:ò%ò)ò)òò#ò%ò4ñ ˜}ð 5ô óJFóôJFr8   r(  rY  c                   ój   ‡ — e Zd ZdZd„ Zd„ Zˆ fd„Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zd„ Zd„ Zd„ Zˆ xZS )Útruncweibull_min_gena9  A doubly truncated Weibull minimum continuous random variable.

    %(before_notes)s

    See Also
    --------
    weibull_min, truncexpon

    Notes
    -----
    The probability density function for `truncweibull_min` is:

    .. math::

        f(x, a, b, c) = \frac{c x^{c-1} \exp(-x^c)}{\exp(-a^c) - \exp(-b^c)}

    for :math:`a < x <= b`, :math:`0 \le a < b` and :math:`c > 0`.

    `truncweibull_min` takes :math:`a`, :math:`b`, and :math:`c` as shape
    parameters.

    Notice that the truncation values, :math:`a` and :math:`b`, are defined in
    standardized form:

    .. math::

        a = (u_l - loc)/scale
        b = (u_r - loc)/scale

    where :math:`u_l` and :math:`u_r` are the specific left and right
    truncation values, respectively. In other words, the support of the
    distribution becomes :math:`(a*scale + loc) < x <= (b*scale + loc)` when
    :math:`loc` and/or :math:`scale` are provided.

    %(after_notes)s

    References
    ----------

    .. [1] Rinne, H. "The Weibull Distribution: A Handbook". CRC Press (2009).

    %(example)s

    c                 ó$   — |dk\  ||kD  z  |dkD  z  S ©Nrˆ   r‡   ©rE   r  r‹   rŒ   s       r6   rc   ztruncweibull_min_gen._argcheckd  s   € Ø�R‘˜A ™EÑ" a¨"¡fÑ-Ð-r8   c                 óÀ   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }t        dddt        j                  fd«      }|||gS )Nr  Fr   r
  r‹   rg   rŒ   rh   )rE   r!  ri  rj  s       r6   rk   z truncweibull_min_gen._shape_infog  sV   € Ü˜˜U Q¬¯© K°Ó@ˆÜ˜˜U Q¬¯© K°Ó?ˆÜ˜˜U Q¬¯© K°Ó@ˆØ�B˜ˆ|Ðr8   c                 ó&   •— t         ‰| �  |d¬«      S )N)r   r   r   rM  ©rA   r•  ©rE   rF   r–  s     €r6   r•  ztruncweibull_min_gen._fitstartm  s   ø€ ä‰wÑ  ¨IÐ Ó6Ð6r8   c                 ó
   — ||fS rN   r‡   rP  s       r6   r–   z!truncweibull_min_gen._get_supportq  ó   € Ø�!ˆtˆr8   c                 óð   — t        j                  t        ||«       «      t        j                  t        ||«       «      z
  }|t        ||dz
  «      z  t        j                  t        ||«       «      z  |z  S r^   ©rP   r·   rÈ  )rE   rq   r  r‹   rŒ   Údenums         r6   rr   ztruncweibull_min_gen._pdft  s]   € Ü—‘œ˜Q ›˜
Ó#¤b§f¡f¬c°!°Q«i¨ZÓ&8Ñ8ˆØ”C˜˜1˜Q™3“K‘¤"§&¡&¬#¨a°«)¨Ó"4Ñ4¸Ñ=Ð=r8   c           	      ó(  — t        j                  t        j                  t        ||«       «      t        j                  t        ||«       «      z
  «      }t        j                  |«      t	        j
                  |dz
  |«      z   t        ||«      z
  |z
  S r^   )rP   rð   r·   rÈ  rw   rx  )rE   rq   r  r‹   rŒ   Úlogdenums         r6   rÞ   ztruncweibull_min_gen._logpdfx  si   € Ü—6‘6œ"Ÿ&™&¤# a¨£) Ó,¬r¯v©v´s¸1¸a³y°jÓ/AÑAÓBˆÜ�v‰v�a‹yœ2Ÿ8™8 A¨¡E¨1Ó-Ñ-´°A°q³	Ñ9¸HÑDÐDr8   c                 ó  — t        j                  t        ||«       «      t        j                  t        ||«       «      z
  }t        j                  t        ||«       «      t        j                  t        ||«       «      z
  }||z  S rN   rX  ©rE   rq   r  r‹   rŒ   r@  rY  s          r6   ru   ztruncweibull_min_gen._cdf|  ód   € Ü�v‰v”s˜1˜a“y�jÓ!¤B§F¡F¬C°°1«I¨:Ó$6Ñ6ˆÜ—‘œ˜Q ›˜
Ó#¤b§f¡f¬c°!°Q«i¨ZÓ&8Ñ8ˆØ�U‰{Ðr8   c           	      ó\  — t        j                  t        j                  t        ||«       «      t        j                  t        ||«       «      z
  «      }t        j                  t        j                  t        ||«       «      t        j                  t        ||«       «      z
  «      }||z
  S rN   ©rP   rð   r·   rÈ  ©rE   rq   r  r‹   rŒ   Úlognumr[  s          r6   rã   ztruncweibull_min_gen._logcdf�  ów   € Ü—‘œŸ™¤ A q£	˜zÓ*¬R¯V©V´S¸¸A³Y°JÓ-?Ñ?Ó@ˆÜ—6‘6œ"Ÿ&™&¤# a¨£) Ó,¬r¯v©v´s¸1¸a³y°jÓ/AÑAÓBˆØ˜Ñ Ð r8   c                 ó  — t        j                  t        ||«       «      t        j                  t        ||«       «      z
  }t        j                  t        ||«       «      t        j                  t        ||«       «      z
  }||z  S rN   rX  r]  s          r6   ry   ztruncweibull_min_gen._sf†  r^  r8   c           	      ó\  — t        j                  t        j                  t        ||«       «      t        j                  t        ||«       «      z
  «      }t        j                  t        j                  t        ||«       «      t        j                  t        ||«       «      z
  «      }||z
  S rN   r`  ra  s          r6   rç   ztruncweibull_min_gen._logsf‹  rc  r8   c                 óØ   — t        t        j                  d|z
  t        j                  t        ||«       «      z  |t        j                  t        ||«       «      z  z   «       d|z  «      S r^   ©rÈ  rP   rð   r·   ©rE   r}   r  r‹   rŒ   s        r6   r�   ztruncweibull_min_gen._isf�  óY   € ÜÜ�V‰V�Q˜‘UœbŸf™f¤c¨!¨Q£i ZÓ0Ñ0°1´r·v±v¼sÀ1Àa»y¸jÓ7IÑ3IÑIÓJÐJÈAÈaÉCóð 	r8   c                 óØ   — t        t        j                  d|z
  t        j                  t        ||«       «      z  |t        j                  t        ||«       «      z  z   «       d|z  «      S r^   rg  rh  s        r6   r~   ztruncweibull_min_gen._ppf•  ri  r8   c           	      ó`  — t        j                  ||z  dz   «      t        j                  ||z  dz   t        ||«      «      t        j                  ||z  dz   t        ||«      «      z
  z  }t	        j
                  t        ||«       «      t	        j
                  t        ||«       «      z
  }||z  S r  )rw   rØ  r½  rÈ  rP   r·   )rE   rb   r  r‹   rŒ   Ú	gamma_funrY  s          r6   r  ztruncweibull_min_gen._munpš  s�   € Ü—H‘H˜Q˜q™S 2™XÓ&Ü�K‰K˜˜!™˜b™¤# a¨£)Ó,¬r¯{©{¸1¸Q¹3À¹8ÄSÈÈAÃYÓ/OÑOñˆ	ô —‘œ˜Q ›˜
Ó#¤b§f¡f¬c°!°Q«i¨ZÓ&8Ñ8ˆØ˜5Ñ Ð r8   )rƒ   r„   r…   r†   rc   rk   r•  r–   rr   rÞ   ru   rã   ry   rç   r�   r~   r  rÐ  rÑ  s   @r6   rM  rM  7  sK   ø„ ñ+òX.òô7òò>òEòò
!ò
ò
!ò
ò
ö
!r8   rM  Útruncweibull_minr®  c                   óF   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zy)Úweibull_max_gena0  Weibull maximum continuous random variable.

    The Weibull Maximum Extreme Value distribution, from extreme value theory
    (Fisher-Gnedenko theorem), is the limiting distribution of rescaled
    maximum of iid random variables. This is the distribution of -X
    if X is from the `weibull_min` function.

    %(before_notes)s

    See Also
    --------
    weibull_min

    Notes
    -----
    The probability density function for `weibull_max` is:

    .. math::

        f(x, c) = c (-x)^{c-1} \exp(-(-x)^c)

    for :math:`x < 0`, :math:`c > 0`.

    `weibull_max` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    https://en.wikipedia.org/wiki/Weibull_distribution

    https://en.wikipedia.org/wiki/Fisher-Tippett-Gnedenko_theorem

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zweibull_max_gen._shape_infoË  r  r8   c                 ól   — |t        | |dz
  «      z  t        j                  t        | |«       «      z  S r^   r+  r
  s      r6   rr   zweibull_max_gen._pdfÎ  s0   € à”�a�R˜˜1™“‰~œbŸf™f¤c¨1¨"¨a£j [Ó1Ñ1Ð1r8   c                 ó~   — t        j                  |«      t        j                  |dz
  | «      z   t	        | |«      z
  S r^   r-  r
  s      r6   rÞ   zweibull_max_gen._logpdfÒ  s3   € Ü�v‰v�a‹yœ2Ÿ8™8 A a¡C¨!¨Ó,Ñ,¬s°A°2°q«zÑ9Ð9r8   c                 óD   — t        j                  t        | |«       «      S rN   rX  r
  s      r6   ru   zweibull_max_gen._cdfÕ  s   € Ü�v‰v”s˜A˜2˜q“z�kÓ"Ð"r8   c                 ó   — t        | |«       S rN   r4  r
  s      r6   rã   zweibull_max_gen._logcdfØ  s   € Ü�Q�B˜“
ˆ{Ðr8   c                 óF   — t        j                  t        | |«       «       S rN   r/  r
  s      r6   ry   zweibull_max_gen._sfÛ  s   € Ü—‘œ#˜q˜b !›*˜Ó%Ð%Ð%r8   c                 óJ   — t        t        j                  |«       d|z  «       S r  )rÈ  rP   rð   r  s      r6   r~   zweibull_max_gen._ppfÞ  s    € Ü”R—V‘V˜A“Y�J  A¡Ó&Ð&Ð&r8   c                 óv   — t        j                  d|dz  |z  z   «      }t        |«      dz  rd}||z  S d}||z  S )Nr‰   rU   r¿  r   )rw   rØ  r  )rE   rb   r  ÚvalÚsgns        r6   r  zweibull_max_gen._munpá  sI   € Ü�h‰h�s˜1˜S™5 ™7‘{Ó#ˆÜˆq‹6�AŠ:ØˆCð �S‰yÐð ˆCØ�S‰yÐr8   c                 óV   — t          |z  t        j                  |«      z
  t         z   dz   S r^   r8  r~  s     r6   rò   zweibull_max_gen._entropyé  r9  r8   N)rƒ   r„   r…   r†   rk   rr   rÞ   ru   rã   ry   r~   r  rò   r‡   r8   r6   ro  ro  ¦  s6   „ ñ#òHEò2ò:ò#òò&ò'òó4r8   ro  Úweibull_max)rŒ   r�   c                   óL   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zy)Úgenlogistic_gena…  A generalized logistic continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genlogistic` is:

    .. math::

        f(x, c) = c \frac{\exp(-x)}
                         {(1 + \exp(-x))^{c+1}}

    for real :math:`x` and :math:`c > 0`. In literature, different
    generalizations of the logistic distribution can be found. This is the type 1
    generalized logistic distribution according to [1]_. It is also referred to
    as the skew-logistic distribution [2]_.

    `genlogistic` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    .. [1] Johnson et al. "Continuous Univariate Distributions", Volume 2,
           Wiley. 1995.
    .. [2] "Generalized Logistic Distribution", Wikipedia,
           https://en.wikipedia.org/wiki/Generalized_logistic_distribution

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zgenlogistic_gen._shape_info  r  r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  r
  s      r6   rr   zgenlogistic_gen._pdf  r½  r8   c                 óà   — |dz
   |dk  z  dz
  }t        j                  |«      }t        j                  |«      ||z  z   |dz   t        j                  t        j
                  | «      «      z  z
  S ©Nr   r   )rP   rŽ  rð   rw   r¦  r·   )rE   rq   r  Úmultr‰  s        r6   rÞ   zgenlogistic_gen._logpdf  sb   € ð �Q‘ˆx˜1˜q™5Ñ! AÑ%ˆÜ�v‰v�a‹yˆÜ�v‰v�a‹y˜4 ™9Ñ$¨¨!©¬r¯x©x¼¿¹À¸u»Ó/FÑ'FÑFÐFr8   c                 ó@   — dt        j                  | «      z   | z  }|S r^   ru  )rE   rq   r  ÚCxs       r6   ru   zgenlogistic_gen._cdf   s!   € Ø”—‘˜�r“
‰l˜q˜bÑ!ˆØˆ	r8   c                 ó\   — | t        j                  t        j                  | «      «      z  S rN   )rP   r¦  r·   r
  s      r6   rã   zgenlogistic_gen._logcdf$  s"   € Øˆr”B—H‘HœRŸV™V Q B›ZÓ(Ñ(Ð(r8   c                 ó\   — t        j                  t        j                  |d|z  «      «       S r?  )rP   rð   rw   Úpowm1r  s      r6   r~   zgenlogistic_gen._ppf'  s#   € Ü—‘”r—x‘x  4¨¡6Ó*Ó+Ð+Ð+r8   c                 óN   — t        j                  | j                  ||«      «       S rN   ©rw   r  rã   r
  s      r6   ry   zgenlogistic_gen._sf*  ó   € Ü—‘˜Ÿ™ a¨Ó+Ó,Ð,Ð,r8   c                 ó,   — | j                  d|z
  |«      S r^   ©r~   r  s      r6   r�   zgenlogistic_gen._isf-  s   € Ø�y‰y˜˜Q™ Ó"Ð"r8   c                 ó¤  — t         t        j                  |«      z   }t        j                  t        j                  z  dz  t        j
                  d|«      z   }dt        j
                  d|«      z  dt        z  z   }|t        j                  |d«      z  }t        j                  dz  dz  dt        j
                  d|«      z  z   }||d	z  z  }||||fS )
Nr�  rU   r  r†  rÊ  r$  ç      .@r…  r¶   )r$   rw   r[  rP   rñ   Úzetar%   rÌ  ©rE   r  rC  rD  rE  rF  s         r6   r   zgenlogistic_gen._stats0  s©   € Ü”b—f‘f˜Q“iÑˆÜ�e‰e”B—E‘E‰k˜#‰o¤§¡¨¨1£Ñ-ˆØ”—‘˜˜1“Ñ ¤&¡Ñ(ˆØ
Œb�h‰h�s˜CÓ Ñ ˆÜ�U‰U�A‰X�d‰]˜QœrŸw™w q¨!›}™_Ñ,ˆØ
ˆc�3‰h‰ˆØ�3˜˜BˆÐr8   c                 ó,   — t        |dk  |fd„ d„ ¬«      S )Ng    €„^Ac                 ót   — t        j                  | «       t        j                  | dz   «      z   t        z   dz   S r^   )rP   rð   rw   r[  r$   rý  s    r6   rç  z*genlogistic_gen._entropy.<locals>.<lambda>;  s+   € ¤R§V¡V¨A£Y J´·±¸¸A¹³Ñ$>ÄÑ$GÈ!Ñ$K€ r8   c                 ó&   — dd| z  z  t         z   dz   S r1  ©r$   rý  s    r6   rç  z*genlogistic_gen._entropy.<locals>.<lambda>A  s   €  q¨!¨a©%¡y´6Ñ'9¸AÑ'=€ r8   rê  rì  r~  s     r6   rò   zgenlogistic_gen._entropy9  s!   € Ü˜!˜c™' A 5ÙKñ >ô?ð 	?r8   N)rƒ   r„   r…   r†   rk   rr   rÞ   ru   rã   r~   ry   r�   r   rò   r‡   r8   r6   r}  r}  ð  s<   „ ñò@Eò*òGòò)ò,ò-ò#òó?r8   r}  Úgenlogisticc                   ó`   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zdd„Zd„ Zd„ Zy)Úgenpareto_gena‰  A generalized Pareto continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genpareto` is:

    .. math::

        f(x, c) = (1 + c x)^{-1 - 1/c}

    defined for :math:`x \ge 0` if :math:`c \ge 0`, and for
    :math:`0 \le x \le -1/c` if :math:`c < 0`.

    `genpareto` takes ``c`` as a shape parameter for :math:`c`.

    For :math:`c=0`, `genpareto` reduces to the exponential
    distribution, `expon`:

    .. math::

        f(x, 0) = \exp(-x)

    For :math:`c=-1`, `genpareto` is uniform on ``[0, 1]``:

    .. math::

        f(x, -1) = 1

    %(after_notes)s

    %(example)s

    c                 ó,   — t        j                  |«      S rN   ©rP   rü   r~  s     r6   rc   zgenpareto_gen._argcheckk  ó   € Ü�{‰{˜1‹~Ðr8   c                 ó^   — t        ddt        j                   t        j                  fd«      gS ©Nr  Fr
  rh   rj   s    r6   rk   zgenpareto_gen._shape_infon  ó%   € Ü˜3 ¬¯©¨´·±Ð'8¸.ÓIÐJÐJr8   c                 óÒ   — t        j                  |«      }t        |dk  |fd„ t         j                  «      }t        j                  |dk\  | j
                  | j
                  «      }||fS )Nr   c                 ó   — d| z  S r?  r‡   rý  s    r6   rç  z,genpareto_gen._get_support.<locals>.<lambda>t  s
   €   q¡€ r8   )rP   rû   r   ri   rO  r‹   )rE   r  rŒ   r‹   s       r6   r–   zgenpareto_gen._get_supportq  sV   € Ü�J‰J�q‹MˆÜ�q˜1‘u˜q˜dÙ(Ü—v‘vóˆô �H‰H�Q˜!‘V˜TŸV™V T§V¡VÓ,ˆØ�!ˆtˆr8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  r
  s      r6   rr   zgenpareto_gen._pdfy  r½  r8   c                 ó8   — t        ||k(  |dk7  z  ||fd„ | «      S )Nr   c                 óB   — t        j                  |dz   || z  «       |z  S r  rf  ©rq   r  s     r6   rç  z'genpareto_gen._logpdf.<locals>.<lambda>  s!   € ¬¯
©
°1°r±6¸1¸Q¹3Ó(?Ð'?À!Ñ'C€ r8   rì  r
  s      r6   rÞ   zgenpareto_gen._logpdf}  s,   € Ü˜1 ™6 a¨1¡fÑ-°°1¨vÙCØ˜"óð 	r8   c                 ó4   — t        j                  | | «       S rN   )rw   Úinv_boxcox1pr
  s      r6   ru   zgenpareto_gen._cdf‚  s   € Ü—‘   Q BÓ'Ð'Ð'r8   c                 ó2   — t        j                  | | «      S rN   )rw   Ú
inv_boxcoxr
  s      r6   ry   zgenpareto_gen._sf…  s   € Ü�}‰}˜a˜R ! Ó$Ð$r8   c                 ó8   — t        ||k(  |dk7  z  ||fd„ | «      S )Nr   c                 ó:   — t        j                  || z  «       |z  S rN   r	  r£  s     r6   rç  z&genpareto_gen._logsf.<locals>.<lambda>Š  s   € ¬¯©°°1±« ~¸Ñ'9€ r8   rì  r
  s      r6   rç   zgenpareto_gen._logsfˆ  s,   € Ü˜1 ™6 a¨1¡fÑ-°°1¨vÙ9Ø˜"óð 	r8   c                 ó4   — t        j                  | | «       S rN   )rw   Úboxcox1pr  s      r6   r~   zgenpareto_gen._ppf�  s   € Ü—‘˜Q˜B  Ó#Ð#Ð#r8   c                 ó2   — t        j                  || «       S rN   )rw   Úboxcoxr  s      r6   r�   zgenpareto_gen._isf�  s   € Ü—	‘	˜!˜a˜RÓ Ð Ð r8   c                 óN  — d|vrd }n!t        |dk  |fd„ t        j                  «      }d|vrd }n!t        |dk  |fd„ t        j                  «      }d|vrd }n!t        |dk  |fd	„ t        j                  «      }d
|vrd }n!t        |dk  |fd„ t        j                  «      }||||fS )Nr?  r   c                 ó   — dd| z
  z  S r^   r‡   ©Úxis    r6   rç  z&genpareto_gen._stats.<locals>.<lambda>˜  s   €  a¨¨R©¡j€ r8   rË  r”   c                 ó*   — dd| z
  dz  z  dd| z  z
  z  S r1  r‡   r°  s    r6   rç  z&genpareto_gen._stats.<locals>.<lambda>ž  s   €  a¨1¨r©6°A©+¡o¸¸Q¸r¹T¹Ñ&B€ r8   r  gUUUUUUÕ?c                 ó\   — dd| z   z  t        j                  dd| z  z
  «      z  dd| z  z
  z  S )NrU   r   r†  r‡  r°  s    r6   rç  z&genpareto_gen._stats.<locals>.<lambda>¤  s4   €  q¨A°©F¡|´b·g±g¸aÀ!ÀBÁ$¹hÓ6GÑ'GØ()¨A¨b©D©ñ(2€ r8   r  rÕ  c                 ó`   — ddd| z  z
  z  d| dz  z  | z   dz   z  dd| z  z
  z  dd| z  z
  z  dz
  S )Nr†  r   rU   r$  r‡   r°  s    r6   rç  z&genpareto_gen._stats.<locals>.<lambda>«  sQ   €  q¨A°°"±©H¡~¸¸2¸q¹5¹À2¹ÈÑ9IÑ'JØ()¨A¨b©D©ñ(2Ø56¸¸2¹±Xñ(?ØABñ(C€ r8   ©r   rP   ri   r  )rE   r  r  r?  rË  r  r  s          r6   r   zgenpareto_gen._stats“  sÉ   € Ø�gÑØ‰Aä˜1˜q™5 1 $Ù0ÜŸ6™6ó#ˆAð �gÑØ‰Aä˜1˜s™7 Q DÙBÜŸ6™6ó#ˆAð �gÑØ‰Aä˜1˜s™7 Q Dñ3äŸ6™6ó#ˆAð �gÑØ‰Aä˜1˜s™7 Q DñDäŸ6™6ó#ˆAð �!�Q˜ˆzÐr8   c           	      ód   ‡‡— d„ Št        |dk7  |fˆˆfd„t        j                  ‰dz   «      «      S )Nc                 ó  — d}t        j                  d| dz   «      }t        |t        j                  | |«      «      D ]  \  }}||d|z  z  d||z  z
  z  z   }Œ t        j
                  || z  dk  |d|z  | z  z  t         j                  «      S )Nrˆ   r   r   r¿  r‰   r<  )rP   rM  Úziprw   ÚcombrO  ri   )rb   r  rx  r  ÚkiÚcnks         r6   rV  z#genpareto_gen._munp.<locals>.__munp±  s‹   € ØˆCÜ—	‘	˜!˜Q ™UÓ#ˆAÜ˜q¤"§'¡'¨!¨Q£-Ó0ò >‘��CØ˜C 2¨"¡*Ñ,°°a¸"±f±Ñ=Ñ=‘ð>ä—8‘8˜A ™E A™I s¨d°Q©h¸1©_Ñ'<¼b¿f¹fÓEÐEr8   r   c                 ó   •—  ‰‰| «      S rN   r‡   )r  Ú_genpareto_gen__munprb   s    €€r6   rç  z%genpareto_gen._munp.<locals>.<lambda>¸  s   ø€ ¡F¨1¨a£L€ r8   r   )r   rw   rØ  )rE   rb   r  r½  s    ` @r6   r  zgenpareto_gen._munp°  s4   ù€ ò	Fô ˜!˜q™& 1 $Ü0ÜŸ(™( 1 q¡5›/ó+ð 	+r8   c                 ó   — d|z   S r  r‡   r~  s     r6   rò   zgenpareto_gen._entropy»  ó   € Ø�A‰vˆr8   Nr  )rƒ   r„   r…   r†   rc   rk   r–   rr   rÞ   ru   ry   rç   r~   r�   r   r  rò   r‡   r8   r6   r—  r—  G  sJ   „ ñ"òFòKòò*òò
(ò%òò
$ò!óò:	+ór8   r—  Ú	genparetoc                   ó:   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
y	)
Úgenexpon_gena!  A generalized exponential continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genexpon` is:

    .. math::

        f(x, a, b, c) = (a + b (1 - \exp(-c x)))
                        \exp(-a x - b x + \frac{b}{c}  (1-\exp(-c x)))

    for :math:`x \ge 0`, :math:`a, b, c > 0`.

    `genexpon` takes :math:`a`, :math:`b` and :math:`c` as shape parameters.

    %(after_notes)s

    References
    ----------
    H.K. Ryu, "An Extension of Marshall and Olkin's Bivariate Exponential
    Distribution", Journal of the American Statistical Association, 1993.

    N. Balakrishnan, Asit P. Basu (editors), *The Exponential Distribution:
    Theory, Methods and Applications*, Gordon and Breach, 1995.
    ISBN 10: 2884491929

    %(example)s

    c                 óÀ   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }t        dddt        j                  fd«      }|||gS )Nr‹   Fr   r
  rŒ   r  rh   )rE   ri  rj  r!  s       r6   rk   zgenexpon_gen._shape_infoâ  sV   € Ü˜˜U Q¬¯© K°Ó@ˆÜ˜˜U Q¬¯© K°Ó@ˆÜ˜˜U Q¬¯© K°Ó@ˆØ�B˜ˆ|Ðr8   c           	      ó¾   — ||t        j                  | |z  «       z  z   t        j                  | |z
  |z  |t        j                  | |z  «       z  |z  z   «      z  S rN   ©rw   r  rP   r·   ©rE   rq   r‹   rŒ   r  s        r6   rr   zgenexpon_gen._pdfè  sh   € ð �AœŸ™ !  A¡›�Ñ'Ñ'¬¯©°!°°A±°q±Ø01´B·H±H¸a¸RÀ¹T³N°?Ñ0CÀAÑ0Eñ1Fó *Gñ Gð 	Gr8   c                 ó¾   — t        j                  ||t        j                  | |z  «       z  z   «      | |z
  |z  z   |t        j                  | |z  «       z  |z  z   S rN   ©rP   rð   rw   r  rÆ  s        r6   rÞ   zgenexpon_gen._logpdfî  sY   € Ü�v‰v�a˜œBŸH™H a R¨¡T›N˜?Ñ+Ñ+Ó,°°°1±°a©xÑ7¸¼B¿H¹HÀaÀRÈÁT»N¸?Ñ8KÈAÑ8MÑMÐMr8   c                 ó~   — t        j                  | |z
  |z  |t        j                  | |z  «       z  |z  z   «       S rN   ry  rÆ  s        r6   ru   zgenexpon_gen._cdfñ  s=   € Ü—‘˜1˜"˜Q™$ ™ A¬¯©°!°°A±« Ñ$7¸Ñ$9Ñ9Ó:Ð:Ð:r8   c                 óÊ   — ||z   }||t        j                  | «      z  z
  |z  }|t        j                  | |z  t        j                  | «      z  «      j
                  z   |z  S rN   )rP   r¦  rw   Úlambertwr·   Úreal©rE   rô  r‹   rŒ   r  r  rÏ  s          r6   r~   zgenexpon_gen._ppfô  s\   € Ø�‰EˆØ�”2—8‘8˜Q˜B“<‘Ñ Ñ"ˆØ”B—K‘K   1¡¤r§v¡v¨q¨b£zÑ 1Ó2×7Ñ7Ñ7¸Ñ:Ð:r8   c                 ó|   — t        j                  | |z
  |z  |t        j                  | |z  «       z  |z  z   «      S rN   rÃ  rÆ  s        r6   ry   zgenexpon_gen._sfù  s:   € Ü�v‰v˜�r˜!‘t˜Q‘h ¤R§X¡X¨q¨b°©d£^ OÑ!4°QÑ!6Ñ6Ó7Ð7r8   c                 óÈ   — ||z   }||t        j                  |«      z  z
  |z  }|t        j                  | |z  t        j                  | «      z  «      j
                  z   |z  S rN   )rP   rð   rw   rË  r·   rÌ  rÍ  s          r6   r�   zgenexpon_gen._isfü  sY   € Ø�‰EˆØ�”2—6‘6˜!“9‘‰_˜aÑˆØ”B—K‘K   1¡¤r§v¡v¨q¨b£zÑ 1Ó2×7Ñ7Ñ7¸Ñ:Ð:r8   Nrµ  r‡   r8   r6   rÂ  rÂ  Â  s,   „ ñò>òGòNò;ò;ò
8ó;r8   rÂ  Úgenexponc                   óv   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zd„ Zd„ Zˆ fd„Zd„ Zd„ Zˆ xZS )Úgenextreme_genaB  A generalized extreme value continuous random variable.

    %(before_notes)s

    See Also
    --------
    gumbel_r

    Notes
    -----
    For :math:`c=0`, `genextreme` is equal to `gumbel_r` with
    probability density function

    .. math::

        f(x) = \exp(-\exp(-x)) \exp(-x),

    where :math:`-\infty < x < \infty`.

    For :math:`c \ne 0`, the probability density function for `genextreme` is:

    .. math::

        f(x, c) = \exp(-(1-c x)^{1/c}) (1-c x)^{1/c-1},

    where :math:`-\infty < x \le 1/c` if :math:`c > 0` and
    :math:`1/c \le x < \infty` if :math:`c < 0`.

    Note that several sources and software packages use the opposite
    convention for the sign of the shape parameter :math:`c`.

    `genextreme` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 ó,   — t        j                  |«      S rN   r™  r~  s     r6   rc   zgenextreme_gen._argcheck,  rš  r8   c                 ó^   — t        ddt        j                   t        j                  fd«      gS rœ  rh   rj   s    r6   rk   zgenextreme_gen._shape_info/  r�  r8   c                 ó  — t        j                  |dkD  dt        j                  |t        «      z  t         j                  «      }t        j                  |dk  dt        j
                  |t         «      z  t         j                   «      }||fS ©Nr   r‰   )rP   rO  Úmaximumr"   ri   Úminimum)rE   r  Ú_bÚ_as       r6   r–   zgenextreme_gen._get_support2  sc   € Ü�X‰X�a˜!‘e˜S¤2§:¡:¨a´Ó#7Ñ7¼¿¹Ó@ˆÜ�X‰X�a˜!‘e˜S¤2§:¡:¨a´%°Ó#8Ñ8¼2¿6¹6¸'ÓBˆØ�2ˆvˆr8   c                 ó8   — t        ||k(  |dk7  z  ||fd„ | «      S )Nr   c                 ó:   — t        j                  | | z  «      |z  S rN   r	  r£  s     r6   rç  z+genextreme_gen._loglogcdf.<locals>.<lambda>:  s   € ¤r§x¡x°°°1±£~°aÑ'7€ r8   rì  r
  s      r6   Ú
_loglogcdfzgenextreme_gen._loglogcdf7  s+   € ä˜1 ™6 a¨1¡fÑ-°°1¨vÙ7¸!¸ó=ð 	=r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  r
  s      r6   rr   zgenextreme_gen._pdf<  s   € ô �v‰v�d—l‘l 1 aÓ(Ó)Ð)r8   c                 óÈ  — t        ||k(  |dk7  z  ||fd„ d«      }t        j                  | «      }| j                  ||«      }t	        j
                  |«      }t	        j                  ||dk(  |t        j                   k(  z  d«       t        |dk(  |t        j                   k(  z   |||fd„ t        j                   ¬«      }t	        j                  ||dk(  |dk(  z  d«       |S )Nr   c                 ó   — || z  S rN   r‡   r£  s     r6   rç  z(genextreme_gen._logpdf.<locals>.<lambda>C  s
   € À!ÀAÁ#€ r8   rˆ   r   c                 ó   — |  |z   |z
  S rN   r‡   )Úpex2Úlpex2Úlex2s      r6   rç  z(genextreme_gen._logpdf.<locals>.<lambda>K  s   € °t°e¸e±mÀdÑ6J€ r8   rÿ  )r   rw   r¦  rÝ  rP   r·   Úputmaskri   )rE   rq   r  ÚcxÚlogex2Úlogpex2râ  Úlogpdfs           r6   rÞ   zgenextreme_gen._logpdfB  sÏ   € Ü˜˜a™ A¨¡FÑ+¨a°¨VÑ5EÀsÓKˆÜ—‘˜2˜#“ˆØ—/‘/ ! QÓ'ˆÜ�v‰v�g‹ˆä
�
‰
�7˜Q !™V¨¬b¯f©f¨W©Ñ5°sÔ;Ü˜r Q™w¨2´"·&±&°©=Ñ9Ð:Ø! 7¨FÐ3ÙJÜ')§v¡v gô/ˆô 	�
‰
�6˜A ™F q¨A¡vÑ.°Ô4Øˆr8   c                 óN   — t        j                  | j                  ||«      «       S rN   )rP   r·   rÝ  r
  s      r6   rã   zgenextreme_gen._logcdfP  s   € Ü—‘�t—‘ q¨!Ó,Ó-Ð-Ð-r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rO  r
  s      r6   ru   zgenextreme_gen._cdfS  r‘  r8   c                 óN   — t        j                  | j                  ||«      «       S rN   r‰  r
  s      r6   ry   zgenextreme_gen._sfV  rŠ  r8   c                 óŠ   — t        j                  t        j                  |«       «       }t        ||k(  |dk7  z  ||fd„ |«      S )Nr   c                 ó<   — t        j                  | | z  «       |z  S rN   ry  r£  s     r6   rç  z%genextreme_gen._ppf.<locals>.<lambda>\  ó   € ¬¯©°!°°a±Ó(8Ð'8¸1Ñ'<€ r8   )rP   rð   r   ©rE   r}   r  rq   s       r6   r~   zgenextreme_gen._ppfY  sF   € Ü�V‰V”R—V‘V˜A“Y�JÓÐˆÜ˜1 ™6 a¨1¡fÑ-°°1¨vÙ<¸aóAð 	Ar8   c                 óŒ   — t        j                  t        j                  | «       «       }t	        ||k(  |dk7  z  ||fd„ |«      S )Nr   c                 ó<   — t        j                  | | z  «       |z  S rN   ry  r£  s     r6   rç  z%genextreme_gen._isf.<locals>.<lambda>a  rï  r8   )rP   rð   rw   r¦  r   rð  s       r6   r�   zgenextreme_gen._isf^  sH   € Ü�V‰V”R—X‘X˜q˜b“\�MÓ"Ð"ˆÜ˜1 ™6 a¨1¡fÑ-°°1¨vÙ<¸aóAð 	Ar8   c                 óN  ‡— ˆfd„} |d«      } |d«      } |d«      } |d«      }t        j                  t        ‰«      dk  ‰t         j                  z  dz  dz  ||dz  z
  «      }d	„ }t	        t        ‰«      dk\  ‰f|t         j                  dz  dz  ¬
«      }	d}
d„ }t	        t        ‰«      |
k\  ‰f|t
         ¬
«      }t        j                  ‰dk  t         j                  | «      }t        j                  ‰dk  t         j                  |dz  |	z  «      }d„ }dt        j                  d«      z  t        z  t         j                  dz  z  }||||f}t	        t        ‰«      |
dz  kD  ‰f|z   ||¬
«      }d„ }|||||f}t	        t        ‰«      |
dz  kD  ‰f|z   |d¬
«      }||||fS )Nc                 ó:   •— t        j                  | ‰z  dz   «      S r^   rj  )rb   r  s    €r6   Úgz genextreme_gen._stats.<locals>.gd  s   ø€ Ü—8‘8˜A ™E A™IÓ&Ð&r8   r   rU   r†  r$  gH¯¼šò×z>r¶   r�  c                 ó¢   — t        j                  t        j                  d| z  dz   «      dt        j                  | dz   «      z  z
  «      | dz  z  S )Nr¶   r‰   rU   ©rw   r  rÆ  rý  s    r6   Úgam2k_fz&genextreme_gen._stats.<locals>.gam2k_fk  sB   € Ü—8‘8œBŸJ™J s¨1¡u¨S¡yÓ1°!´B·J±J¸qÀ3¹wÓ4GÑ2GÑGÓHÈÈCÉÑOÐOr8   ©r�  r   ç›+¡†›„=c                 ó^   — t        j                  t        j                  | dz   «      «      | z  S r^   r÷  rý  s    r6   Úgamk_fz%genextreme_gen._stats.<locals>.gamk_fo  s#   € Ü—8‘8œBŸJ™J q¨1¡uÓ-Ó.¨qÑ0Ð0r8   r<  r$  c                 óP   — d„ }t        | dk\  | f|z   |t        j                  ¬«      S )Nc                 óX   — t        j                  | «      | |d|z  z   |z  z   z  |dz  z  S ©NrU   rÊ  rO   )r  rE  rF  Úg3Úg2mg12s        r6   Ú
sk1_eval_fz;genextreme_gen._stats.<locals>.sk1_eval.<locals>.sk1_eval_f{  s2   € Ü—w‘w˜q“z B 3¨"¨q°©x©-¸Ñ);Ñ#;Ñ<¸VÀS¹[ÑHÐHr8   rò  rù  rm  )r  rG   r  s      r6   Úsk1_evalz'genextreme_gen._stats.<locals>.sk1_evalz  s'   € òIä˜a 5™j¨1¨$¨t©)°zÌRÏVÉVÔTÐTr8   rÀ  r…  g�Âõ(\�Ò?c                 óH   — d„ }t        | dk\  ||t        j                  ¬«      S )Nc                 óB   — |d|z  d||z   z  | z  z   | z  z   |dz  z  dz
  S )NrÕ  r†  rU   r‡   )rE  rF  r   Úg4r  s        r6   Ú
ku1_eval_fz;genextreme_gen._stats.<locals>.ku1_eval.<locals>.ku1_eval_f…  s6   € Ø˜b ™e a¨¨f©¡o°bÑ&8Ñ8¸"Ñ<Ñ<¸fÀa¹iÑGÈ!ÑKÐKr8   g      Ð¿rÿ  rm  )r  rG   r  s      r6   Úku1_evalz'genextreme_gen._stats.<locals>.ku1_eval„  s!   € òLä˜a 5™j¨$°
ÄbÇfÁfÔMÐMr8   gq=
×£pÍ?ç333333@)	rP   rO  rŽ  rñ   r   r$   r  rÿ   r%   )rE   r  rõ  rE  rF  r   r  r  rø  Úgam2kÚepsrü  Úgamkr?  rË  r  Úsk_fillrG   r‘  r  r’  s    `                   r6   r   zgenextreme_gen._statsc  s™  ø€ ô	'áˆq‹TˆÙˆq‹TˆÙˆq‹TˆÙˆq‹TˆÜ—‘œ#˜a›& 4™-¨!¬B¯E©E©'°C©¸Ñ);¸RÀÀCÁ¹ZÓHˆò	Päœ3˜q›6 T™>¨A¨4°7ÄbÇeÁeÈSÁjÐQTÁnÔUˆØˆò	1äœ#˜a›& C™-¨!¨°ÄFÀ7ÔKˆô �H‰H�Q˜‘XœrŸv™v¨ uÓ-ˆô �H‰H�Q˜‘XœrŸv™v r¨3¡w¨u¡}Ó5ˆò	Uð
 ”R—W‘W˜Q“Z‘-¤Ñ&¤r§u¡u¨a¡xÑ/ˆØ�B˜˜FÐ#ˆÜœ˜A›  d¡Ñ*¨Q¨D°©I¸ÈWÔUˆò	Nð
 �B˜˜B Ð'ˆÜœ˜A›  d¡Ñ*¨Q¨D°©I¸ÈXÔVˆà�!�R˜ˆ|Ðr8   c                 ó’   •— t        |t        «      r|j                  «       }t        |«      }|dk  rd}nd}t        ‰| �  ||f¬«      S )Nr   r”   r$  rM  ©r?   r*   r“  r   rA   r•  )rE   rF   rõ  r‹   r–  s       €r6   r•  zgenextreme_gen._fitstartŽ  sJ   ø€ Ü�dœLÔ)Ø—>‘>Ó#ˆDä�$‹KˆØˆqŠ5Ø‰AàˆAÜ‰wÑ  ¨Q¨DÐ Ó1Ð1r8   c                 ó6  — t        j                  d|dz   «      }d||z  z  t        j                  t        j                  ||«      d|z  z  t        j
                  ||z  dz   «      z  d¬«      z  }t        j                  ||z  dkD  |t         j                  «      S )Nr   r   r‰   r¿  rú  )rP   rM  r¥  rw   r¹  rØ  rO  ri   )rE   rb   r  r  Úvalss        r6   r  zgenextreme_gen._munp™  sƒ   € Ü�I‰I�a˜˜1™ÓˆØ�1�a‘4‰xœ"Ÿ&™&Ü�G‰G�A�q‹M˜R !™GÑ#¤b§h¡h¨q°©s°Q©wÓ&7Ñ7Øôñ ˆô �x‰x˜˜!™˜b™ $¬¯©Ó/Ð/r8   c                 ó    — t         d|z
  z  dz   S r^   r”  r~  s     r6   rò   zgenextreme_gen._entropy   s   € Ü�q˜1‘u‰~ Ñ!Ð!r8   )rƒ   r„   r…   r†   rc   rk   r–   rÝ  rr   rÞ   rã   ru   ry   r~   r�   r   r•  r  rò   rÐ  rÑ  s   @r6   rÒ  rÒ    sX   ø„ ñ%òLòKòò
=ò
*òò.ò*ò-òAò
Aò
)ôV	2ò0ö"r8   rÒ  Ú
genextremec                 óJ  ‡ — d}ˆ fd„}‰ dkD  r7t        j                  ‰ «      dz   }‰ dk  rDt        j                  ||d¬«      }|S ‰ dkD  rt        j                  ‰ d	z  «      d
z   }n	d‰  |z
  z  }t        j                  ||dd¬«      \  }}}}|dk7  rt        d‰ ›�«      ‚|d   S )af  Inverse of the digamma function (real positive arguments only).

    This function is used in the `fit` method of `gamma_gen`.
    The function uses either optimize.fsolve or optimize.newton
    to solve `sc.digamma(x) - y = 0`.  There is probably room for
    improvement, but currently it works over a wide range of y:

    >>> import numpy as np
    >>> rng = np.random.default_rng()
    >>> y = 64*rng.standard_normal(1000000)
    >>> y.min(), y.max()
    (-311.43592651416662, 351.77388222276869)
    >>> x = [_digammainv(t) for t in y]
    >>> np.abs(sc.digamma(x) - y).max()
    1.1368683772161603e-13

    g¶oüŒxâ?c                 ó4   •— t        j                  | «      ‰z
  S rN   )rw   rÐ  r£  s    €r6   r^  z_digammainv.<locals>.func»  s   ø€ Ü�z‰z˜!‹}˜qÑ Ð r8   g      À¿r”   r¶  ç»½×Ùß|Û=)ÚtolrÔ  g-²�ï§@gë­�­,¶?r‰   ç•dyáý¥=T)Úxtolr   r   z _digammainv: fsolve failed, y = r   )rP   r·   r   Únewtonr”  ÚRuntimeError)r�  Ú_emr^  Úx0Úvaluer©  rª  rW  s   `       r6   Ú_digammainvr  §  s¹   ø€ ð$ &€Cô!ð 	ˆ6‚zÜ�V‰V�A‹Y˜‰_ˆØˆrŠ6ô —O‘O D¨"°%Ô8ˆEØˆLØ	
ˆRŠÜ�V‰V�A�e‘G‹_˜wÑ&‰à�Q�B˜‘HÑˆä%Ÿ_™_¨T°2¸EØ9=ô?Ñ€Eˆ4��dà
ˆa‚xÜÐ=¸a¸UÐCÓDÐDà�‰8€Or8   c                   óŠ   ‡ — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zd„ Zˆ fd„Z eed¬«      ˆ fd„«       Zˆ xZS )Ú	gamma_gena“  A gamma continuous random variable.

    %(before_notes)s

    See Also
    --------
    erlang, expon

    Notes
    -----
    The probability density function for `gamma` is:

    .. math::

        f(x, a) = \frac{x^{a-1} e^{-x}}{\Gamma(a)}

    for :math:`x \ge 0`, :math:`a > 0`. Here :math:`\Gamma(a)` refers to the
    gamma function.

    `gamma` takes ``a`` as a shape parameter for :math:`a`.

    When :math:`a` is an integer, `gamma` reduces to the Erlang
    distribution, and when :math:`a=1` to the exponential distribution.

    Gamma distributions are sometimes parameterized with two variables,
    with a probability density function of:

    .. math::

        f(x, \alpha, \beta) =
        \frac{\beta^\alpha x^{\alpha - 1} e^{-\beta x }}{\Gamma(\alpha)}

    Note that this parameterization is equivalent to the above, with
    ``scale = 1 / beta``.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r	  rh   rj   s    r6   rk   zgamma_gen._shape_info  r  r8   c                 ó&   — |j                  ||«      S rN   ©Ústandard_gamma)rE   r‹   r×   rØ   s       r6   rÙ   zgamma_gen._rvs  s   € Ø×*Ñ*¨1¨dÓ3Ð3r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  r  s      r6   rr   zgamma_gen._pdf  r½  r8   c                 óf   — t        j                  |dz
  |«      |z
  t        j                  |«      z
  S r  )rw   rx  rÆ  r  s      r6   rÞ   zgamma_gen._logpdf  s)   € Ü�x‰x˜˜#™˜qÓ! AÑ%¬¯
©
°1«Ñ5Ð5r8   c                 ó.   — t        j                  ||«      S rN   r¼  r  s      r6   ru   zgamma_gen._cdf  r‚   r8   c                 ó.   — t        j                  ||«      S rN   r¿  r  s      r6   ry   zgamma_gen._sf  s   € Ü�|‰|˜A˜qÓ!Ð!r8   c                 ó.   — t        j                  ||«      S rN   rë  r  s      r6   r~   zgamma_gen._ppf  s   € Ü�~‰~˜a Ó#Ð#r8   c                 ó.   — t        j                  ||«      S rN   ©rw   rÈ  r  s      r6   r�   zgamma_gen._isf  s   € Ü�‰˜q !Ó$Ð$r8   c                 ó@   — ||dt        j                  |«      z  d|z  fS )Nr¶   r�  r‡  r  s     r6   r   zgamma_gen._stats  s!   € Ø�!�SœŸ™ ›‘^ S¨¡UÐ*Ð*r8   c                 ó.   — t        j                  ||«      S rN   ©rw   rË  ©rE   rb   r‹   s      r6   r  zgamma_gen._munp  s   € Ü�w‰w�q˜!‹}Ðr8   c                 ó4   — d„ }d„ }t        |dk  |f||¬«      S )Nc                 ój   — t        j                  | «      d| z
  z  | z   t        j                  | «      z   S r^   ©rw   r[  rÆ  ©r‹   s    r6   rÒ  z+gamma_gen._entropy.<locals>.regular_formula#  s+   € Ü—6‘6˜!“9  !¡Ñ$ qÑ(¬2¯:©:°a«=Ñ8Ð8r8   c                 óÎ   — ddt        j                  dt         j                  z  «      z   t        j                  | «      z   z  dd| z  z  z
  | dz  dz  z
  | dz  d	z  z
  | d
z  dz  z   S )Nr”   r‰   rU   r   r†  r³  rÀ  r´  r  rµ  r·  rï   r4  s    r6   r×  z.gamma_gen._entropy.<locals>.asymptotic_formula&  sq   € ð
 ˜2¤§¡ q¬¯©¡w£Ñ/´"·&±&¸³)Ñ;Ñ<¸qÀ!ÀaÁ%¹yÑHØ˜#‘v˜r‘kñ"Ø%&¨¡V¨R¡Kñ0Ø34°c±6¸3±,ñ?ð @r8   éú   rê  rì  )rE   r‹   rÒ  r×  s       r6   rò   zgamma_gen._entropy!  s+   € ò	9ò	@ô ˜!˜c™' A 5¨/Ø/ô1ð 	1r8   c                 ó”   •— t        |t        «      r|j                  «       }t        |«      }dd|dz  z   z  }t        ‰| �  ||f¬«      S )Nr$  ç:Œ0âŽyE>rU   rM  r  )rE   rF   r‘  r‹   r–  s       €r6   r•  zgamma_gen._fitstart1  sM   ø€ ô �dœLÔ)Ø—>‘>Ó#ˆDÜ�4‹[ˆØ�˜˜A™‘ÑˆÜ‰wÑ  ¨Q¨DÐ Ó1Ð1r8   a<          When the location is fixed by using the argument `floc`
        and `method='MLE'`, this
        function uses explicit formulas or solves a simpler numerical
        problem than the full ML optimization problem.  So in that case,
        the `optimizer`, `loc` and `scale` arguments are ignored.
        

ró   c                 óÊ  •‡— |j                  dd «      }|j                  dd«      }t        |t        «      s|€&|j                  «       dk7  rt	        ‰| �  |g|¢­i |¤ŽS |j                  dd «       t        |g d¢«      }|j                  dd «      }t        |«       |�|�|�t        d«      ‚t        j                  |«      }t        j                  |«      j                  «       st        d«      ‚|j                  «       dk(  r°t        j                  |«      }t        j                  |«      }	t        j                  ||z
  d	z  «      }
|||}}}|€|€
|€|
d
|	z  z  }|€|€t        j                   |	|z  «      }|€
|€|	||z
  z  }|€
|€|	|d
z  z  }|€||z
  |z  }|€|||z  z
  }|€||z
  |z  }|||fS t        j"                  ||k  «      rt%        d|t        j&                  ¬«      ‚|dk7  r||z
  }|j                  «       }|€—|�|}nŒt        j(                  |«      t        j(                  |«      j                  «       z
  Šd	‰z
  t        j                   ‰d	z
  d
z  d‰z  z   «      z   d‰z  z  }|dz  }|dz  }t+        j,                  ˆfd„||d¬«      }||z  }nFt        j(                  |«      j                  «       t        j(                  |«      z
  }t/        |«      }|}|||fS )Nrö   r1   r;   r<   r˜  r÷   rø   rù   r†  rU   rØ  rŸ  r   r  rÀ  g333333ã?gffffffö?c                 ó`   •— t        j                  | «      t        j                  | «      z
  ‰z
  S rN   )rP   rð   rw   rÐ  )r‹   r  s    €r6   rç  zgamma_gen.fit.<locals>.<lambda>›  s!   ø€ ¬b¯f©f°Q«i¼"¿*¹*ÀQ»-Ñ.GÈ!Ñ.K€ r8   )Údisp)r=   r?   r*   r>   rA   rC   r3   r   r7   rú   rP   rû   rü   rý   rþ   r§  rÿ   r£  rK  ri   rð   r   Úbrentqr  )rE   rF   rG   r5   rö   r1   r™  r÷   Úm1Úm2Úm3r‹   r.   r/   r¨  ÚaestÚxarÀ  r  r  r–  s                      @€r6   rC   zgamma_gen.fit=  só  ù€ ð �x‰x˜ Ó%ˆØ—‘˜( EÓ*ˆä�tœ\Ô*Ø� §¡£°4Ò!7ô ‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ð 	�‰�˜Ôä! $Ò(=Ó>ˆØ—‘˜( DÓ)ˆä$ TÔ*àˆ>˜dÐ.°6Ð3Eô ð )ó *ð *ô �z‰z˜$Óˆä�{‰{˜4Ó ×$Ñ$Ô&ÜÐCÓDÐDð �<‰<‹>˜TÒ!Ü—‘˜“ˆBÜ—‘˜“ˆBÜ—‘˜$ ™)¨Ñ)Ó*ˆBØ  f�EˆsˆAàˆy˜S˜[¨U¨]Ø˜a "™f™�àˆ{˜u˜}ÜŸ™  Q¡›�Øˆy˜U˜]Ø˜b 3™h™�Øˆy˜S˜[Ø˜% 1™*Ñ%�àˆyØ˜#‘X Ñ&�Øˆ{Ø˜1˜u™9‘n�Øˆ}Ø˜c™ Q™�Ø�c˜5�=Ð ô
 �6‰6�$˜$‘,ÔÜ˜w¨d¼"¿&¹&ÔAÐAà�1Š9ð ˜$‘;ˆDØ�y‰y‹{ˆð ˆ>àˆ~à‘ô —F‘F˜4“L¤2§6¡6¨$£<×#4Ñ#4Ó#6Ñ6�Ø˜!™œbŸg™g q¨¡s¨Q¡h°°A±¡oÓ6Ñ6¸2¸a¹4Ñ@�Ø˜5‘\�Ø˜5‘\�Ü—O‘OÓ$KØ$&¨°ô4�ð
 ˜1‘H‰Eô
 —‘�t“×!Ñ!Ó#¤b§f¡f¨V£nÑ4ˆAÜ˜A“ˆAØˆEà�$˜ˆ~Ðr8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   ry   r~   r�   r   r  rò   r•  r	   r   rC   rÐ  rÑ  s   @r6   r!  r!  Ù  sh   ø„ ñ'òPEó4ò*ò6ò!ò"ò$ò%ò+òò1ô 
2ñ ˜}ð 5ô óeóôer8   r!  rØ  c                   óR   ‡ — e Zd ZdZd„ Zd„ Zˆ fd„Z eed¬«      ˆ fd„«       Z	ˆ xZ
S )Ú
erlang_gena¿  An Erlang continuous random variable.

    %(before_notes)s

    See Also
    --------
    gamma

    Notes
    -----
    The Erlang distribution is a special case of the Gamma distribution, with
    the shape parameter `a` an integer.  Note that this restriction is not
    enforced by `erlang`. It will, however, generate a warning the first time
    a non-integer value is used for the shape parameter.

    Refer to `gamma` for examples.

    c                 óª   — t        j                  t        j                  |«      |k(  «      }|s"d|›d�}t        j                  |t
        d¬«       |dkD  S )NzRThe shape parameter of the erlang distribution has been given a non-integer value r2   r†  ©Ú
stacklevelr   )rP   rý   ÚfloorÚwarningsÚwarnÚRuntimeWarning)rE   r‹   ÚallintÚmessages       r6   rc   zerlang_gen._argcheckÃ  sM   € Ü—‘œŸ™ › qÑ(Ó)ˆÙð=Ø=>¸EÀðDˆGä�M‰M˜'¤>¸aÕ@Ø�1‰uˆr8   c                 ó@   — t        dddt        j                  fd«      gS )Nr‹   Tr   rg   rh   rj   s    r6   rk   zerlang_gen._shape_infoÍ  rl   r8   c                 óª   •— t        |t        «      r|j                  «       }t        ddt	        |«      dz  z   z  «      }t
        t        | �  ||f¬«      S )NrJ  r8  rU   rM  )r?   r*   r“  r  r   rA   r!  r•  )rE   rF   r‹   r–  s      €r6   r•  zerlang_gen._fitstartÐ  sP   ø€ ô �dœLÔ)Ø—>‘>Ó#ˆDÜ��tœe D›k¨1™nÑ,Ñ-Ó.ˆÜ”Y Ñ/°¸A¸4Ð/Ó@Ð@r8   a¦          The Erlang distribution is generally defined to have integer values
        for the shape parameter.  This is not enforced by the `erlang` class.
        When fitting the distribution, it will generally return a non-integer
        value for the shape parameter.  By using the keyword argument
        `f0=<integer>`, the fit method can be constrained to fit the data to
        a specific integer shape parameter.ró   c                 ó*   •— t        ‰| �  |g|¢­i |¤ŽS rN   )rA   rC   ©rE   rF   rG   r5   r–  s       €r6   rC   zerlang_gen.fitÛ  s   ø€ ô ‰w‰{˜4Ð/ $Ò/¨$Ñ/Ð/r8   )rƒ   r„   r…   r†   rc   rk   r•  r	   r   rC   rÐ  rÑ  s   @r6   rC  rC  ¯  s9   ø„ ñò&òCôAñ ˜}ð 5/ô 0ó0ó0ô0r8   rC  Úerlangc                   óT   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zdd„Z	d	„ Z
d
„ Zd„ Zd„ Zd„ Zy)Úgengamma_gena½  A generalized gamma continuous random variable.

    %(before_notes)s

    See Also
    --------
    gamma, invgamma, weibull_min

    Notes
    -----
    The probability density function for `gengamma` is ([1]_):

    .. math::

        f(x, a, c) = \frac{|c| x^{c a-1} \exp(-x^c)}{\Gamma(a)}

    for :math:`x \ge 0`, :math:`a > 0`, and :math:`c \ne 0`.
    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `gengamma` takes :math:`a` and :math:`c` as shape parameters.

    %(after_notes)s

    References
    ----------
    .. [1] E.W. Stacy, "A Generalization of the Gamma Distribution",
       Annals of Mathematical Statistics, Vol 33(3), pp. 1187--1192.

    %(example)s

    c                 ó   — |dkD  |dk7  z  S r2  r‡   )rE   r‹   r  s      r6   rc   zgengamma_gen._argcheck	  ó   € Ø�A‘˜!˜q™&Ñ!Ð!r8   c                 ó    — t        dddt        j                  fd«      }t        ddt        j                   t        j                  fd«      }||gS r¨  rh   r©  s      r6   rk   zgengamma_gen._shape_info  óB   € Ü˜˜U Q¬¯© K°Ó@ˆÜ˜˜U¤b§f¡f W¬b¯f©fÐ$5°~ÓFˆØ�Bˆxˆr8   c                 óN   — t        j                  | j                  |||«      «      S rN   rÝ  r«  s       r6   rr   zgengamma_gen._pdf  ó   € Ü�v‰v�d—l‘l 1 a¨Ó+Ó,Ð,r8   c                 ó\   ‡— t        |dk7  |dkD  z  ||fˆfd„t        j                   ¬«      S )Nr   c                 ó²   •— t        j                  t        |«      «      t        j                  |‰z  dz
  | «      z   | |z  z
  t        j
                  ‰«      z
  S r^   )rP   rð   rŽ  rw   rx  rÆ  )rq   r  r‹   s     €r6   rç  z&gengamma_gen._logpdf.<locals>.<lambda>  sG   ø€ ¬¯©¬s°1«v«¼¿¹À!ÀAÁ#ÈÁ'È1Ó9MÑ(MØ*+¨Q©$ñ)/Ü13·±¸A³ñ)?€ r8   rÿ  r  r«  s     ` r6   rÞ   zgengamma_gen._logpdf  s4   ø€ Ü˜1 ™6 a¨!¡eÑ,¨q°!¨fó@ä%'§V¡V Gô-ð 	-r8   c                 ó˜   — ||z  }t        j                  ||«      }t        j                  ||«      }t        j                  |dkD  ||«      S r2  ©rw   r½  rÀ  rP   rO  ©rE   rq   r‹   r  ÚxcÚval1Úval2s          r6   ru   zgengamma_gen._cdf  óB   € Ø�‰TˆÜ�{‰{˜1˜bÓ!ˆÜ�|‰|˜A˜rÓ"ˆÜ�x‰x˜˜A™˜t TÓ*Ð*r8   Nc                 ó8   — |j                  ||¬«      }|d|z  z  S )Nr  r‰   r$  )rE   r‹   r  r×   rØ   rõ  s         r6   rÙ   zgengamma_gen._rvs   s%   € Ø×'Ñ'¨°Ð'Ó5ˆØ�2�a‘4‰yÐr8   c                 ó˜   — ||z  }t        j                  ||«      }t        j                  ||«      }t        j                  |dkD  ||«      S r2  r]  r^  s          r6   ry   zgengamma_gen._sf$  rb  r8   c                 óš   — t        j                  ||«      }t        j                  ||«      }t        j                  |dkD  ||«      d|z  z  S rÖ  ©rw   rÄ  rÈ  rP   rO  ©rE   r}   r‹   r  r`  ra  s         r6   r~   zgengamma_gen._ppf*  óB   € Ü�~‰~˜a Ó#ˆÜ�‰˜q !Ó$ˆÜ�x‰x˜˜A™˜t TÓ*¨S°©UÑ3Ð3r8   c                 óš   — t        j                  ||«      }t        j                  ||«      }t        j                  |dkD  ||«      d|z  z  S rÖ  rf  rg  s         r6   r�   zgengamma_gen._isf/  rh  r8   c                 ó:   — t        j                  ||dz  |z  «      S r  r/  )rE   rb   r‹   r  s       r6   r  zgengamma_gen._munp4  s   € ä�w‰w�q˜!˜C™% ™'Ó"Ð"r8   c                 ó:   — d„ }d„ }t        |dk\  ||f||¬«      }|S )Nc                 ó¾   — t        j                  | «      }| d|z
  z  ||z  z   }t        j                  | «      t        j                  t        |«      «      z
  }||z   }|S r^   )rw   r[  rÆ  rP   rð   rŽ  )r‹   r  rx  ÚAÚBró  s         r6   r¯  z&gengamma_gen._entropy.<locals>.regular9  sQ   € Ü—&‘&˜“)ˆCØ�Q˜‘W‘  a¡Ñ'ˆAÜ—
‘
˜1“¤§¡¤s¨1£v£Ñ.ˆAØ�A‘ˆAØˆHr8   c                 ó:  — t         j                  «       t        j                  | «      dz  z
  t        j                  t        j                  |«      «      z
  | dz  dz  z   | dz  dz  z
  t        j                  | «      | dz  dz  z
  | dz  dz  z
  | dz  d	z  z   |z  z   S )
NrU   r<  r…  r´  r  r³  rÀ  rµ  r·  )r  rò   rP   rð   rŽ  )r‹   r  s     r6   Ú
asymptoticz)gengamma_gen._entropy.<locals>.asymptotic@  s™   € ä—M‘M“O¤b§f¡f¨Q£i°¡kÑ1Ü—f‘fœRŸV™V A›YÓ'ñ(Ø+,¨c©6°1©*ñ5Ø89¸3¹À±{ñCä—v‘v˜a“y A s¡F¨A¡:Ñ-°°C±¸±Ñ;¸qÀ#¹vÀs¹lÑJÈAÑMñNð Or8   ç      i@rŒ  rì  )rE   r‹   r  r¯  rp  ró  s         r6   rò   zgengamma_gen._entropy8  s,   € ò	ò	Oô �q˜C‘x ! Q ¨:¸'ÔBˆØˆr8   r  )rƒ   r„   r…   r†   rc   rk   rr   rÞ   ru   rÙ   ry   r~   r�   r  rò   r‡   r8   r6   rS  rS  é  s>   „ ñò>"òò
-ò-ò+óò+ò4ò
4ò
#ór8   rS  Úgengammac                   ó4   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	y)	Úgenhalflogistic_gena¡  A generalized half-logistic continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `genhalflogistic` is:

    .. math::

        f(x, c) = \frac{2 (1 - c x)^{1/(c-1)}}{[1 + (1 - c x)^{1/c}]^2}

    for :math:`0 \le x \le 1/c`, and :math:`c > 0`.

    `genhalflogistic` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zgenhalflogistic_gen._shape_infoc  r  r8   c                 ó$   — | j                   d|z  fS r  r4  r~  s     r6   r–   z genhalflogistic_gen._get_supportf  s   € Ø�v‰v�s˜1‘uˆ}Ðr8   c                 óx   — d|z  }t        j                  d||z  z
  «      }||dz
  z  }||z  }d|z  d|z   dz  z  S rí  ©rP   rû   )rE   rq   r  ÚlimitrØ  Útmp0Útmp2s          r6   rr   zgenhalflogistic_gen._pdfi  sP   € ð �A‘ˆÜ�j‰j˜˜1˜Q™3™ÓˆØ�U˜1‘W‰~ˆØ�C‰xˆØ�‰v˜˜4™ !™Ñ#Ð#r8   c                 ób   — d|z  }t        j                  d||z  z
  «      }||z  }d|z
  d|z   z  S r)  rx  )rE   rq   r  ry  rØ  r{  s         r6   ru   zgenhalflogistic_gen._cdfr  s=   € Ø�A‘ˆÜ�j‰j˜˜1˜Q™3™ÓˆØ�U‰|ˆØ�D‘˜Q˜t™VÑ$Ð$r8   c                 ó0   — d|z  dd|z
  d|z   z  |z  z
  z  S r)  r‡   r  s      r6   r~   zgenhalflogistic_gen._ppfx  s'   € Ø�1‰u�a˜#˜a™% # a¡%™¨1Ñ,Ñ,Ñ-Ð-r8   c                 óD   — dd|z  dz   t        j                  d«      z  z
  S r‹  r2  r~  s     r6   rò   zgenhalflogistic_gen._entropy{  s"   € Ø�A�a‘C˜‘Eœ2Ÿ6™6 !›9Ñ$Ñ$Ð$r8   N)
rƒ   r„   r…   r†   rk   r–   rr   ru   r~   rò   r‡   r8   r6   rt  rt  M  s&   „ ñò*Eòò$ò%ò.ó%r8   rt  Úgenhalflogisticc                   óp   ‡ — e Zd ZdZd„ Zd„ Zˆ fd„Zd„ Zd„ Zd„ e	d„ «       «       Z
d	„ Zd
„ Zdd„Zd„ Zˆ xZS )Úgenhyperbolic_genu  A generalized hyperbolic continuous random variable.

    %(before_notes)s

    See Also
    --------
    t, norminvgauss, geninvgauss, laplace, cauchy

    Notes
    -----
    The probability density function for `genhyperbolic` is:

    .. math::

        f(x, p, a, b) =
            \frac{(a^2 - b^2)^{p/2}}
            {\sqrt{2\pi}a^{p-1/2}
            K_p\Big(\sqrt{a^2 - b^2}\Big)}
            e^{bx} \times \frac{K_{p - 1/2}
            (a \sqrt{1 + x^2})}
            {(\sqrt{1 + x^2})^{1/2 - p}}

    for :math:`x, p \in ( - \infty; \infty)`,
    :math:`|b| < a` if :math:`p \ge 0`,
    :math:`|b| \le a` if :math:`p < 0`.
    :math:`K_{p}(.)` denotes the modified Bessel function of the second
    kind and order :math:`p` (`scipy.special.kv`)

    `genhyperbolic` takes ``p`` as a tail parameter,
    ``a`` as a shape parameter,
    ``b`` as a skewness parameter.

    %(after_notes)s

    The original parameterization of the Generalized Hyperbolic Distribution
    is found in [1]_ as follows

    .. math::

        f(x, \lambda, \alpha, \beta, \delta, \mu) =
           \frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)}
           e^{\beta (x - \mu)} \times \frac{K_{\lambda - 1/2}
           (\alpha \sqrt{\delta^2 + (x - \mu)^2})}
           {(\sqrt{\delta^2 + (x - \mu)^2} / \alpha)^{1/2 - \lambda}}

    for :math:`x \in ( - \infty; \infty)`,
    :math:`\gamma := \sqrt{\alpha^2 - \beta^2}`,
    :math:`\lambda, \mu \in ( - \infty; \infty)`,
    :math:`\delta \ge 0, |\beta| < \alpha` if :math:`\lambda \ge 0`,
    :math:`\delta > 0, |\beta| \le \alpha` if :math:`\lambda < 0`.

    The location-scale-based parameterization implemented in
    SciPy is based on [2]_, where :math:`a = \alpha\delta`,
    :math:`b = \beta\delta`, :math:`p = \lambda`,
    :math:`scale=\delta` and :math:`loc=\mu`

    Moments are implemented based on [3]_ and [4]_.

    For the distributions that are a special case such as Student's t,
    it is not recommended to rely on the implementation of genhyperbolic.
    To avoid potential numerical problems and for performance reasons,
    the methods of the specific distributions should be used.

    References
    ----------
    .. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions
       on Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
       pp. 151-157, 1978. https://www.jstor.org/stable/4615705

    .. [2] Eberlein E., Prause K. (2002) The Generalized Hyperbolic Model:
        Financial Derivatives and Risk Measures. In: Geman H., Madan D.,
        Pliska S.R., Vorst T. (eds) Mathematical Finance - Bachelier
        Congress 2000. Springer Finance. Springer, Berlin, Heidelberg.
        :doi:`10.1007/978-3-662-12429-1_12`

    .. [3] Scott, David J, WÃ¼rtz, Diethelm, Dong, Christine and Tran,
       Thanh Tam, (2009), Moments of the generalized hyperbolic
       distribution, MPRA Paper, University Library of Munich, Germany,
       https://EconPapers.repec.org/RePEc:pra:mprapa:19081.

    .. [4] E. Eberlein and E. A. von Hammerstein. Generalized hyperbolic
       and inverse Gaussian distributions: Limiting cases and approximation
       of processes. FDM Preprint 80, April 2003. University of Freiburg.
       https://freidok.uni-freiburg.de/fedora/objects/freidok:7974/datastreams/FILE1/content

    %(example)s

    c                 óÀ   — t        j                  t        j                  |«      |k  |dk\  «      t        j                  t        j                  |«      |k  |dk  «      z  S r2  )rP   Úlogical_andrŽ  )rE   rô  r‹   rŒ   s       r6   rc   zgenhyperbolic_gen._argcheckÜ  sH   € Ü—‘œrŸv™v a›y¨1™}¨a°1©fÓ5Ü—.‘.¤§¡¨£¨a¡°°Q±Ó7ñ8ð 	9r8   c                 óü   — t        ddt        j                   t        j                  fd«      }t        dddt        j                  fd«      }t        ddt        j                   t        j                  fd«      }|||gS )Nrô  Fr
  r‹   r   rg   rŒ   rh   )rE   Úipri  rj  s       r6   rk   zgenhyperbolic_gen._shape_infoà  sd   € Ü˜˜U¤b§f¡f W¬b¯f©fÐ$5°~ÓFˆÜ˜˜U Q¬¯© K°Ó?ˆÜ˜˜U¤b§f¡f W¬b¯f©fÐ$5°~ÓFˆØ�B˜ˆ|Ðr8   c                 ó&   •— t         ‰| �  |d¬«      S )N)r   r   r”   rM  rS  rT  s     €r6   r•  zgenhyperbolic_gen._fitstartæ  s   ø€ ô ‰wÑ  ¨KÐ Ó8Ð8r8   c                 óD   — t         j                  d„ «       } |||||«      S )Nc                 ó2   — t        j                  | |||«      S rN   )r   Úgenhyperbolic_logpdf©rq   rô  r‹   rŒ   s       r6   Ú_logpdf_singlez1genhyperbolic_gen._logpdf.<locals>._logpdf_singleî  s   € ä×.Ñ.¨q°!°Q¸Ó:Ð:r8   ©rP   Ú	vectorize)rE   rq   rô  r‹   rŒ   r‹  s         r6   rÞ   zgenhyperbolic_gen._logpdfë  s-   € ô 
�‰ñ	;ó 
ð	;ñ ˜a  A qÓ)Ð)r8   c                 óD   — t         j                  d„ «       } |||||«      S )Nc                 ó2   — t        j                  | |||«      S rN   )r   Úgenhyperbolic_pdfrŠ  s       r6   Ú_pdf_singlez+genhyperbolic_gen._pdf.<locals>._pdf_single÷  s   € ä×+Ñ+¨A¨q°!°QÓ7Ð7r8   rŒ  )rE   rq   rô  r‹   rŒ   r‘  s         r6   rr   zgenhyperbolic_gen._pdfô  s-   € ô 
�‰ñ	8ó 
ð	8ñ ˜1˜a  AÓ&Ð&r8   c                 óN   — t        j                  | t         j                  g¬«      S )N©Úotypes©rP   r�  Úfloat64)r^  s    r6   rç  zgenhyperbolic_gen.<lambda>   s   € ”"—,‘,˜t¬R¯Z©Z¨LÔ9€ r8   c                 óÐ  — t        j                  |||gt        «      j                  j	                  t        j
                  «      }t        j                  t        d|«      }t        j                  ||z   ||z
  z  «      }||z  t        j                  |dz   |«      z  t        j                  ||«      z  }d}	d}
| |cxk  r|k  r?n n<t        j                  || ||	|
¬«      d   t        j                  ||||	|
¬«      d   z   }nt        j                  || ||	|
¬«      d   }t        j                  |«      rd}t        j                   |t"        d¬«       t%        d	t'        d
|«      «      S )zÈ
        Integrate the pdf of the genhyberbolic distribution from x0 to x1.
        This is a private function used by _cdf() and _sf() only; either x0
        will be -inf or x1 will be inf.
        Ú_genhyperbolic_pdfr   r  r   )ÚepsrelÚepsabszdInfinite values encountered in scipy.special.kve. Values replaced by NaN to avoid incorrect results.r†  rE  rˆ   r‰   )rP   ÚarrayrR  ÚctypesÚdata_asÚc_void_pr   Úfrom_cythonr   rÿ   rw   Úkvr   ÚquadÚisnanrH  rI  rJ  Úmaxr‡  )r  r@  rô  r‹   rŒ   Ú	user_dataÚllcrÎ  rþ   r™  rš  ÚintgrlrY   s                r6   Ú_integrate_pdfz genhyperbolic_gen._integrate_pdf   sJ  € ô —H‘H˜a  A˜Y¬Ó.×5Ñ5×=Ñ=¼f¿o¹oÓNˆ	Ü×*Ñ*¬6Ð3GØ+4ó6ˆä�G‰G�Q˜‘U˜Q ™U‘OÓ$ˆØ�‰s”R—U‘U˜1˜q™5 !“_Ñ$¤r§u¡u¨Q°£{Ñ2ˆØˆØˆØ�Œ>�r�>ô  —n‘n S¨"¨dØ,2¸6ôCØCDñFä!Ÿ™ s¨D°"Ø.4¸VôEØEFñHñH‰Fô
 —^‘^ C¨¨RØ+1¸&ôBØBCñEˆFä�8‰8�FÔðHˆCä�M‰M˜#œ~¸!Õ<Ü�3œ˜C Ó(Ó)Ð)r8   c                 óJ   — | j                  t        j                   ||||«      S rN   ©r§  rP   ri   ©rE   rq   rô  r‹   rŒ   s        r6   ru   zgenhyperbolic_gen._cdf"  s!   € Ø×"Ñ"¤B§F¡F 7¨A¨q°!°QÓ7Ð7r8   c                 óH   — | j                  |t        j                  |||«      S rN   r©  rª  s        r6   ry   zgenhyperbolic_gen._sf%  s   € Ø×"Ñ" 1¤b§f¡f¨a°°AÓ6Ð6r8   c                 óR  — t        j                  |d«      t        j                  |d«      z
  }t        j                  |d«      }t        j                  |d«      }t        j                  |||||¬«      }	t        j                  ||¬«      }
||	z  t        j
                  |	«      |
z  z   S )NrU   r”   r$  )rô  rŒ   r/   r×   rØ   r×  )rP   Úfloat_powerÚgeninvgaussrÙ  r  rÿ   )rE   rô  r‹   rŒ   r×   rØ   rº  r»  r¼  ÚgigÚnormsts              r6   rÙ   zgenhyperbolic_gen._rvs(  s–   € ô
 �^‰^˜A˜qÓ!¤B§N¡N°1°aÓ$8Ñ8ˆä�^‰^˜B Ó$ˆä�^‰^˜B Ó&ˆÜ�o‰oØØØØØ%ð ó ˆô —‘˜t°,�Ó?ˆà�3‰wœŸ™ ›¨Ñ.Ñ.Ð.r8   c                 ó¸  ‡— t        j                  |||«      \  }}}t        j                  |d«      t        j                  |d«      z
  }t        j                  |d«      }t        j                  dd«      t        j                  |d«      z  }t        j                  ddd«      }|j	                  |j
                  d|j                  z  z   «      }t        j                  ||z   |«      \  Š}}}	}
ˆfd	„|||	|
fD «       \  }}}}||z  |z  }||z  t        j                  |d«      t        j                  |d«      z  |t        j                  |d«      z
  z  z   }t        j                  |d
«      t        j                  |d
«      z  |d
|z  |z  t        j                  ‰d«      z  z
  dt        j                  |d
«      z  z   z  d
|z  t        j                  |d«      z  |t        j                  |d«      z
  z  z   }|t        j                  |d«      z  }t        j                  |d«      t        j                  |d«      z  |d|	z  |z  t        j                  ‰d«      z  z
  d|z  t        j                  |d«      z  t        j                  ‰d«      z  z   d
t        j                  |d«      z  z
  z  t        j                  |d«      t        j                  |d
«      z  d|z  d|z  |z  t        j                  ‰d«      z  z
  dt        j                  |d
«      z  z   z  z   d
t        j                  |d«      z  |z  z   }|t        j                  |d«      z  d
z
  }||||fS )NrU   r”   r   r¿  r   r$  rC  )r   c              3   ó(   •K  — | ]	  }|‰z  –— Œ y ­wrN   r‡   )Ú.0rŒ   Úb0s     €r6   ú	<genexpr>z+genhyperbolic_gen._stats.<locals>.<genexpr>I  s   øè ø€ Ò; Q˜!˜b�&Ñ;ùs   ƒr†  r  rA  r…  rÔ  rÀ  )	rP   rñ  r­  ÚlinspacerN  Úshaper+  rw   r   )rE   rô  r‹   rŒ   rº  r»  ÚintegersÚb1Úb2Úb3Úb4Úr1Úr2Úr3Úr4r?  rË  Úm3er  Úm4er  r´  s                        @r6   r   zgenhyperbolic_gen._stats=  s  ø€ ô ×%Ñ% a¨¨AÓ.‰ˆˆ1ˆaÜ�^‰^˜A˜qÓ!¤B§N¡N°1°aÓ$8Ñ8ˆÜ�^‰^˜B Ó$ˆÜ�^‰^˜A˜qÓ!¤B§N¡N°2°sÓ$;Ñ;ˆÜ—;‘;˜q ! QÓ'ˆà×#Ñ# H§N¡N°T¸A¿F¹F±]Ñ$BÓCˆÜŸU™U 1 x¡<°Ó4ÑˆˆB��B˜Û;¨2¨r°2°rÐ*:Ô;‰ˆˆB��Bà�‰F�R‰Kˆà�‰G”b—n‘n Q¨Ó*¬R¯^©^¸BÀÓ-BÑBØ”"—.‘.  QÓ'Ñ'ñ)ñ )ð 	
ô
 �N‰N˜1˜aÓ ¤2§>¡>°"°aÓ#8Ñ8Ø�!�b‘&˜2‘+¤§¡¨r°2Ó 6Ñ6Ñ6Ø”—‘  AÓ&Ñ&ñ'ñ(ð �‰E”B—N‘N 2 qÓ)Ñ)Ø”"—.‘.  QÓ'Ñ'ñ)ñ)ð 	ð ”"—.‘.  GÓ,Ñ,ˆä�N‰N˜1˜aÓ ¤2§>¡>°"°aÓ#8Ñ8Ø�!�b‘&˜2‘+¤§¡¨r°3Ó 7Ñ7Ñ7Ø�‰V”b—n‘n R¨Ó+Ñ+¬b¯n©n¸RÀÓ.EÑEñFà”—‘  AÓ&Ñ&ñ'ñ(ô �N‰N˜1˜aÓ ¤2§>¡>°"°aÓ#8Ñ8Ø�‰V�b˜2‘g ‘l¤R§^¡^°B¸Ó%<Ñ<Ñ<Ø”—‘  AÓ&Ñ&ñ'ñ(ñ	(ð ”—‘˜r 1Ó%Ñ%¨Ñ*ñ+ð 	ð ”"—.‘.  BÓ'Ñ'¨!Ñ+ˆà�!�Q˜ˆzÐr8   r  )rƒ   r„   r…   r†   rc   rk   r•  rÞ   rr   Ústaticmethodr§  ru   ry   rÙ   r   rÐ  rÑ  s   @r6   r�  r�  ‚  sT   ø„ ñWòr9òô9ò
*ò'ñ :Øñ*ó ó :ð*ò@8ò7ó/ö*'r8   r�  Úgenhyperbolicc                   ó@   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zy
)Úgompertz_genaq  A Gompertz (or truncated Gumbel) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `gompertz` is:

    .. math::

        f(x, c) = c \exp(x) \exp(-c (e^x-1))

    for :math:`x \ge 0`, :math:`c > 0`.

    `gompertz` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zgompertz_gen._shape_info€  r  r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  r
  s      r6   rr   zgompertz_gen._pdfƒ  r½  r8   c                 ód   — t        j                  |«      |z   |t        j                  |«      z  z
  S rN   rÈ  r
  s      r6   rÞ   zgompertz_gen._logpdf‡  s%   € Ü�v‰v�a‹y˜1‰}˜q¤2§8¡8¨A£;™Ñ.Ð.r8   c                 ó\   — t        j                  | t        j                  |«      z  «       S rN   ry  r
  s      r6   ru   zgompertz_gen._cdfŠ  s#   € Ü—‘˜!˜œbŸh™h q›kÑ)Ó*Ð*Ð*r8   c                 ó`   — t        j                  d|z  t        j                  | «      z  «      S r?  r	  r  s      r6   r~   zgompertz_gen._ppf�  s$   € Ü�x‰x˜˜q™¤2§8¡8¨Q¨B£<Ñ/Ó0Ð0r8   c                 óZ   — t        j                  | t        j                  |«      z  «      S rN   rÃ  r
  s      r6   ry   zgompertz_gen._sf�  s    € Ü�v‰v�q�bœ2Ÿ8™8 A›;Ñ&Ó'Ð'r8   c                 óZ   — t        j                  t        j                  |«       |z  «      S rN   rÅ  ©rE   rô  r  s      r6   r�   zgompertz_gen._isf“  s   € Ü�x‰xœŸ™ ›˜
 1™Ó%Ð%r8   c                 óx   — dt        j                  |«      z
  t        j                  j	                  |«      |z  z
  S r  )rP   rð   rw   Ú_ufuncsÚ_scaled_exp1r~  s     r6   rò   zgompertz_gen._entropy–  s-   € Ø”R—V‘V˜A“Y‰¤§¡×!8Ñ!8¸Ó!;¸AÑ!=Ñ=Ð=r8   N©rƒ   r„   r…   r†   rk   rr   rÞ   ru   r~   ry   r�   rò   r‡   r8   r6   rÆ  rÆ  j  s0   „ ñò*Eò*ò/ò+ò1ò(ò&ó>r8   rÆ  Úgompertzc                 óÔ   — t        j                  | «      } t        j                  |«      }|j                  «       }t        j                  ||z
  «      }t        j                  | |¬«      S )N)Úweights)rP   rû   r£  r·   Úaverage)rq   Ú
logweightsÚmaxlogwrÕ  s       r6   Ú_average_with_log_weightsrÙ  �  sM   € Ü
�
‰
�1‹€AÜ—‘˜JÓ'€JØ�n‰nÓ€GÜ�f‰f�Z 'Ñ)Ó*€GÜ�:‰:�a Ô)Ð)r8   c                   ór   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Ze ee«      d„ «       «       Zy)Úgumbel_r_gena  A right-skewed Gumbel continuous random variable.

    %(before_notes)s

    See Also
    --------
    gumbel_l, gompertz, genextreme

    Notes
    -----
    The probability density function for `gumbel_r` is:

    .. math::

        f(x) = \exp(-(x + e^{-x}))

    for real :math:`x`.

    The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
    distribution.  It is also related to the extreme value distribution,
    log-Weibull and Gompertz distributions.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zgumbel_r_gen._shape_infoÁ  r¦   r8   c                 óJ   — t        j                  | j                  |«      «      S rN   rÝ  r©   s     r6   rr   zgumbel_r_gen._pdfÄ  ó   € ä�v‰v�d—l‘l 1“oÓ&Ð&r8   c                 ó6   — | t        j                  | «      z
  S rN   ru  r©   s     r6   rÞ   zgumbel_r_gen._logpdfÈ  s   € Øˆr”B—F‘F˜A˜2“J‰Ðr8   c                 óV   — t        j                  t        j                  | «       «      S rN   ru  r©   s     r6   ru   zgumbel_r_gen._cdfË  s   € Ü�v‰v”r—v‘v˜q˜b“z�kÓ"Ð"r8   c                 ó0   — t        j                  | «       S rN   ru  r©   s     r6   rã   zgumbel_r_gen._logcdfÎ  s   € Ü—‘˜�r“
ˆ{Ðr8   c                 óV   — t        j                  t        j                  |«       «       S rN   r2  r°   s     r6   r~   zgumbel_r_gen._ppfÑ  s   € Ü—‘œŸ™˜q›	�zÓ"Ð"Ð"r8   c                 óX   — t        j                  t        j                  | «       «       S rN   rÅ  r©   s     r6   ry   zgumbel_r_gen._sfÔ  s    € Ü—‘œ"Ÿ&™& ! ›*˜Ó%Ð%Ð%r8   c                 óX   — t        j                  t        j                  | «       «       S rN   ©rP   rð   r¦  r  s     r6   r�   zgumbel_r_gen._isf×  s    € Ü—‘œŸ™ ! ›�}Ó%Ð%Ð%r8   c                 ó¼   — t         t        j                  t        j                  z  dz  dt        j                  d«      z  t        j                  dz  z  t        z  dfS )Nr�  rÀ  r…  r†  r	  ©r$   rP   rñ   rÿ   r%   rj   s    r6   r   zgumbel_r_gen._statsÚ  s?   € Ü”r—u‘uœRŸU™U‘{ 3‘¨¬2¯7©7°1«:©´b·e±e¸Q±hÑ(>ÄÑ(GÈÐOÐOr8   c                 ó   — t         dz   S r  r”  rj   s    r6   rò   zgumbel_r_gen._entropyÝ  s   € ä˜‰{Ðr8   c                 óÂ  ‡‡‡— t        | ‰||«      \  Š}}ˆfd„}|�|} ||«      Š‰|fS |�	|Šˆˆfd„Šnˆfd„Š|j                  dd«      }|dz  |dz  }
}	ˆfd„} ||	|
«      sE|	dkD  s|
t        j                  k  r-|	dz  }	|
dz  }
 ||	|
«      s|	dkD  rŒ|
t        j                  k  rŒ-t	        j
                  ‰|	|
fd	d	¬
«      }|j                  }|�|n ||«      Š‰|fS )Nc                 ó|   •— |  t        j                  ‰ | z  «      t        j                  t	        ‰«      «      z
  z  S rN   )rw   rE  rP   rð   r¤  )r/   rF   s    €r6   Úget_loc_from_scalez,gumbel_r_gen.fit.<locals>.get_loc_from_scaleî  s1   ø€ Ø�6œRŸ\™\¨4¨%°%©-Ó8¼2¿6¹6Ä#ÀdÃ)Ó;LÑLÑMÐMr8   c                 ó”   •— ‰‰z
  t        j                  ‰‰z
  | z  «      z  ‰z   }t        ‰«      ‰| z   z  }|j                  «       |z
  S rN   )rP   r·   r¤  r¥  )r/   Úterm1Úterm2rF   r.   s      €€r6   r^  zgumbel_r_gen.fit.<locals>.func  sK   ø€ Ø  4™Z¬2¯6©6°3¸±:ÀÑ2FÓ+GÑGÈ$ÑN�EÜ ›I¨¨u©Ñ5�EØ Ÿ9™9›;¨Ñ.Ð.r8   c                 óV   •— ‰ | z  }t        ‰|¬«      }‰j                  «       |z
  | z
  S )N)r×  )rÙ  rþ   )r/   ÚsdataÚwavgrF   s      €r6   r^  zgumbel_r_gen.fit.<locals>.func  s0   ø€ Ø!˜E E™M�EÜ4°TÀeÔL�DØŸ9™9›;¨Ñ-°Ñ5Ð5r8   r/   r   rU   c                 ór   •— t        j                   ‰| «      «      t        j                   ‰|«      «      k7  S rN   rO   )rR   rS   r^  s     €r6   rT   z0gumbel_r_gen.fit.<locals>.interval_contains_root  s-   ø€ äŸ™¡ V£Ó-ÜŸ™¡ V£Ó-ñ.ð /r8   r   rú  )rE  Úrtolr  )rG  r=   rP   ri   r   r+   rH  )rE   rF   rG   r5   rö   r÷   rë  r/   Úbrack_startrR   rS   rT   Úresr^  r.   s    `           @@r6   rC   zgumbel_r_gen.fitá  s  ú€ ô 9¸¸tØ9=¸tóEÑˆˆd�Fô	Nð Ðð ˆEÙ$ UÓ+ˆCð\ �EˆzÐðU ÐØ�ö/ô6ð Ÿ(™( 7¨AÓ.ˆKØ(¨1™_¨k¸A©o�FˆFô
/ñ .¨f°fÔ=Ø š
 f¬r¯v©v¢oØ˜!‘�Ø˜!‘�ñ .¨f°fÔ=Ø ›
 f¬r¯v©v£oô ×&Ñ& t°f¸fÐ5EØ,1¸ô?ˆCà—H‘HˆEØÐ*‘$Ñ0BÀ5Ó0IˆCØ�EˆzÐr8   N)rƒ   r„   r…   r†   rk   rr   rÞ   ru   rã   r~   ry   r�   r   rò   rK   r   r   rC   r‡   r8   r6   rÛ  rÛ  ¥  s]   „ ñò6ò'òò#òò#ò&ò&òPòð Ù˜MÓ*ñ@ó +ó ñ@r8   rÛ  Úgumbel_rc                   ór   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Ze ee«      d„ «       «       Zy)Úgumbel_l_gena	  A left-skewed Gumbel continuous random variable.

    %(before_notes)s

    See Also
    --------
    gumbel_r, gompertz, genextreme

    Notes
    -----
    The probability density function for `gumbel_l` is:

    .. math::

        f(x) = \exp(x - e^x)

    for real :math:`x`.

    The Gumbel distribution is sometimes referred to as a type I Fisher-Tippett
    distribution.  It is also related to the extreme value distribution,
    log-Weibull and Gompertz distributions.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zgumbel_l_gen._shape_infoF  r¦   r8   c                 óJ   — t        j                  | j                  |«      «      S rN   rÝ  r©   s     r6   rr   zgumbel_l_gen._pdfI  rÞ  r8   c                 ó2   — |t        j                  |«      z
  S rN   ru  r©   s     r6   rÞ   zgumbel_l_gen._logpdfM  r€  r8   c                 óV   — t        j                  t        j                  |«       «       S rN   rÅ  r©   s     r6   ru   zgumbel_l_gen._cdfP  s   € Ü—‘œ"Ÿ&™& ›)˜Ó$Ð$Ð$r8   c                 óV   — t        j                  t        j                  | «       «      S rN   ©rP   rð   rw   r¦  r°   s     r6   r~   zgumbel_l_gen._ppfS  s   € Ü�v‰v”r—x‘x  “|�mÓ$Ð$r8   c                 ó.   — t        j                  |«       S rN   ru  r©   s     r6   rç   zgumbel_l_gen._logsfV  r  r8   c                 óT   — t        j                  t        j                  |«       «      S rN   ru  r©   s     r6   ry   zgumbel_l_gen._sfY  ó   € Ü�v‰v”r—v‘v˜a“y�jÓ!Ð!r8   c                 óT   — t        j                  t        j                  |«       «      S rN   r2  r©   s     r6   r�   zgumbel_l_gen._isf\  r  r8   c                 ó¾   — t          t        j                  t        j                  z  dz  dt        j                  d«      z  t        j                  dz  z  t        z  dfS )Nr�  éôÿÿÿr…  r†  r	  rç  rj   s    r6   r   zgumbel_l_gen._stats_  sF   € ÜˆwœŸ™œbŸe™e™ C™Ø”2—7‘7˜1“:‰~œbŸe™e Q™hÑ&¬Ñ/°ð8ð 	8r8   c                 ó   — t         dz   S r  r”  rj   s    r6   rò   zgumbel_l_gen._entropyc  s   € Ü˜‰{Ðr8   c                 ó    — |j                  d«      �	|d    |d<   t        j                  t        j                  |«       g|¢­i |¤Ž\  }}| |fS )Nrö   )r=   rö  rC   rP   rû   )rE   rF   rG   r5   Úloc_rÚscale_rs         r6   rC   zgumbel_l_gen.fitf  sV   € ð �8‰8�FÓÐ'Ø  ™L˜=ˆD�‰LÜ"Ÿ,™,¬¯
©
°4Ó(8Ð'8ÐH¸4ÒHÀ4ÑH‰ˆˆwØˆv�wˆÐr8   N)rƒ   r„   r…   r†   rk   rr   rÞ   ru   r~   rç   ry   r�   r   rò   rK   r   r   rC   r‡   r8   r6   rø  rø  )  sZ   „ ñò8ò'òò%ò%òò"ò"ò8òð Ù˜MÓ*ñó +ó ñr8   rø  Úgumbel_lc                   óx   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Ze ee«      ˆ fd„«       «       Zˆ xZS )Úhalfcauchy_gena  A Half-Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `halfcauchy` is:

    .. math::

        f(x) = \frac{2}{\pi (1 + x^2)}

    for :math:`x \ge 0`.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zhalfcauchy_gen._shape_infoŽ  r¦   r8   c                 ó:   — dt         j                  z  d||z  z   z  S r\  r/  r©   s     r6   rr   zhalfcauchy_gen._pdf‘  s   € à”2—5‘5‰y˜#˜a ™c™'Ñ"Ð"r8   c                 ó€   — t        j                  dt         j                  z  «      t        j                  ||z  «      z
  S r>  ©rP   rð   rñ   rw   r¦  r©   s     r6   rÞ   zhalfcauchy_gen._logpdf•  s*   € Ü�v‰v�cœ"Ÿ%™%‘iÓ ¤2§8¡8¨A¨a©C£=Ñ0Ð0r8   c                 óT   — dt         j                  z  t        j                  |«      z  S r>  rï  r©   s     r6   ru   zhalfcauchy_gen._cdf˜  s   € Ø”2—5‘5‰yœŸ™ 1›Ñ%Ð%r8   c                 óT   — t        j                  t         j                  dz  |z  «      S r  ©rP   Útanrñ   r°   s     r6   r~   zhalfcauchy_gen._ppf›  s   € Ü�v‰v”b—e‘e˜A‘g˜a‘iÓ Ð r8   c                 óV   — dt         j                  z  t        j                  d|«      z  S ©Nr¶   r   )rP   rñ   r’  r©   s     r6   ry   zhalfcauchy_gen._sfž  s    € Ø”2—5‘5‰yœ2Ÿ:™: a¨Ó+Ñ+Ð+r8   c                 óZ   — dt        j                  t         j                  |z  dz  «      z  S r  r  r  s     r6   r�   zhalfcauchy_gen._isf¡  s"   € Ø”2—6‘6œ"Ÿ%™% ™' !™)Ó$Ñ$Ð$r8   c                 ó~   — t         j                  t         j                  t         j                  t         j                  fS rN   r  rj   s    r6   r   zhalfcauchy_gen._stats¤  rœ  r8   c                 óN   — t        j                  dt         j                  z  «      S r  rï   rj   s    r6   rò   zhalfcauchy_gen._entropy§  rž  r8   c                 ó  •— |j                  dd«      rt        ‰
| �  |g|¢­i |¤ŽS t        | |||«      \  }}}t	        j
                  |«      }|�$||k  rt        d|t        j                  ¬«      ‚|}n|}d„ }|�|}	||	fS  |||«      }	||	fS )Nr;  FÚ
halfcauchyrŸ  c                 óþ   ‡‡— || z
  }|j                   Št        j                  |«      Šˆˆfd„}t        j                  d«      j                  dz  }t        ||t        j                  |«      f¬«      }|j                  S )Nc                 óP   •— | dz  ‰z   }dt        j                  ‰|z  «      z  ‰z
  S r  ©rP   r¥  )r/   Údenominatorrb   Úshifted_data_squareds     €€r6   Úfun_to_solvez<halfcauchy_gen.fit.<locals>.find_scale.<locals>.fun_to_solveÄ  s1   ø€ Ø# Q™hÐ)=Ñ=�Øœ2Ÿ6™6Ð"6°{Ñ"BÓCÑCÀaÑGÐGr8   r‰   r”   ©rE  )r×   rP   ÚsquareÚfinfoÚtinyr+   r£  rH  )r.   rF   Úshifted_datar   Úsmallrõ  rb   r  s         @@r6   Ú
find_scalez&halfcauchy_gen.fit.<locals>.find_scale¿  sg   ù€ Ø #™:ˆLØ—	‘	ˆAÜ#%§9¡9¨\Ó#:Ð õHô —H‘H˜S“M×&Ñ&¨Ñ+ˆEÜ˜l°U¼B¿F¹FÀ<Ó<PÐ4QÔRˆCØ—8‘8ˆOr8   ©r3   rA   rC   rG  rP   r‡  rK  ri   )rE   rF   rG   r5   rö   r÷   rˆ  r.   r'  r/   r–  s             €r6   rC   zhalfcauchy_gen.fitª  s¸   ø€ ð �8‰8�J Ô&Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ä8¸¸tØ9=¸tóEÑˆˆd�Fô —6‘6˜$“<ˆØÐØ˜$Šä" <°tÄ2Ç6Á6ÔJÐJØ‰Cð ˆCò	ð ÐØˆEð �EˆzÐñ ˜s DÓ)ˆEà�EˆzÐr8   )rƒ   r„   r…   r†   rk   rr   rÞ   ru   r~   ry   r�   r   rò   rK   r   r   rC   rÐ  rÑ  s   @r6   r  r  z  sV   ø„ ñò&ò#ò1ò&ò!ò,ò%ò.òð Ù˜MÓ*ó%ó +ó ô%r8   r  r  c                   óx   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Ze ee«      ˆ fd„«       «       Zˆ xZS )Úhalflogistic_genaÿ  A half-logistic continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `halflogistic` is:

    .. math::

        f(x) = \frac{ 2 e^{-x} }{ (1+e^{-x})^2 }
             = \frac{1}{2} \text{sech}(x/2)^2

    for :math:`x \ge 0`.

    %(after_notes)s

    References
    ----------
    .. [1] Asgharzadeh et al (2011). "Comparisons of Methods of Estimation for the
           Half-Logistic Distribution". Selcuk J. Appl. Math. 93-108.

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zhalflogistic_gen._shape_infoñ  r¦   r8   c                 óJ   — t        j                  | j                  |«      «      S rN   rÝ  r©   s     r6   rr   zhalflogistic_gen._pdfô  s   € ô �v‰v�d—l‘l 1“oÓ&Ð&r8   c                 óŒ   — t        j                  d«      |z
  dt        j                  t        j                  | «      «      z  z
  S rµ   )rP   rð   rw   r¦  r·   r©   s     r6   rÞ   zhalflogistic_gen._logpdfù  s1   € Ü�v‰v�a‹y˜1‰}˜r¤B§H¡H¬R¯V©V°Q°B«ZÓ$8Ñ8Ñ8Ð8r8   c                 ó2   — t        j                  |dz  «      S r>  )rP   Útanhr©   s     r6   ru   zhalflogistic_gen._cdfü  s   € Ü�w‰w�q˜‘u‹~Ðr8   c                 ó2   — dt        j                  |«      z  S r  ©rP   Úarctanhr°   s     r6   r~   zhalflogistic_gen._ppfÿ  s   € Ø”—‘˜A“‰Ðr8   c                 ó4   — dt        j                  | «      z  S r  ©rw   Úexpitr©   s     r6   ry   zhalflogistic_gen._sf  s   € Ø”2—8‘8˜Q˜B“<ÑÐr8   c                 ó,   — t        |dk  |fd„ d„ ¬«      S )Nr”   c                 ó4   — t        j                  d| z  «       S r  ©rw   ÚlogitrÆ   s    r6   rç  z'halflogistic_gen._isf.<locals>.<lambda>  s   € ¤R§X¡X¨c°A©gÓ%6Ð$6€ r8   c                 ó8   — dt        j                  d| z
  «      z  S r‹  r1  rÆ   s    r6   rç  z'halflogistic_gen._isf.<locals>.<lambda>  s   €  q¬¯©°A¸±EÓ):Ñ':€ r8   rê  rì  r°   s     r6   r�   zhalflogistic_gen._isf  s   € Ü˜!˜c™' A 5Ù6Ù:ô<ð 	<r8   c                 ó|  — |dk(  ry|dk(  rdt        j                  d«      z  S |dk(  r$t         j                  t         j                  z  dz  S |dk(  r	dt        z  S |dk(  rdt         j                  dz  z  d	z  S ddt	        d
d|z
  «      z
  z  t        j                  |dz   «      z  t        j                  |d«      z  S )Nr   r   rU   rD  r†  r  r$  rÉ  rŽ  r¶   )rP   rð   rñ   r%   rÈ  rw   rØ  r�  ra   s     r6   r  zhalflogistic_gen._munp
  s©   € Ø�Š6ØØ�Š6Ø”R—V‘V˜A“Y‘;ÐØ�Š6Ü—5‘5œŸ™‘;˜s‘?Ð"Ø�Š6Ø”V‘8ˆOØ�Š6Ø”R—U‘U˜A‘X‘: Ñ$Ð$Ø�!”C˜˜Q˜q™S“M‘/Ñ"¤2§8¡8¨A¨a©C£=Ñ0´·±¸¸A³Ñ>Ð>r8   c                 ó2   — dt        j                  d«      z
  S r  r2  rj   s    r6   rò   zhalflogistic_gen._entropy  r3  r8   c                 ó  •— |j                  dd«      rt        ‰
| �  |g|¢­i |¤ŽS t        | |||«      \  }}}d„ }t	        j
                  |«      }|�$||k  rt        d|t        j                  ¬«      ‚|}n|}|�|n |||«      }	||	fS )Nr;  Fc                 ó²  — | j                   d   }t        j                  | d¬«      }t        j                  d|dz   «      |dz   z  }d|z
  }d|z   }|d|z  |z  t        j                  ||z  «      z  z
  }d|z  |z  }||z
  }dt        j
                  |dd  |dd  z  «      z  }	dt        j
                  |dd  |dd  dz  z  «      z  }
|	t        j                  |	dz  d|z  |
z  z   «      z   d|z  z  }d}d}|j                  «       }||kD  rN|t        j                  | |z  «      z  }|d|z  |j                  «       z  z
  }t        ||z
  |z  «      }|}||kD  rŒN|S )	Nr   rú  r   r”   rU   r.  r$  r8  )r·  rP   ÚsortrM  rð   r¥  rÿ   rþ   rw   r5  rŽ  )rF   r.   Ún_observationsÚsorted_datarô  r}   Úpp1r  rn  rn  ÚCr/   ró  Úrelative_residualÚshifted_meanÚsum_termÚ	scale_news                    r6   r'  z(halflogistic_gen.fit.<locals>.find_scale#  s�  € ð "ŸZ™Z¨™]ˆNÜŸ'™' $¨QÔ/ˆKÜ—	‘	˜!˜^¨aÑ/Ó0°.À1Ñ2DÑEˆAØ�A‘ˆAØ�a‘%ˆCØ˜˜a™ #™¬¯©¨s°Q©w«Ñ7Ñ7ˆEØ˜‘7˜S‘=ˆDØ%¨Ñ+ˆKØ”B—F‘F˜5  ˜9 {°1°2 Ñ6Ó7Ñ7ˆAØ”B—F‘F˜4  ˜8 k°!°" o°qÑ&8Ñ8Ó9Ñ9ˆAàœ"Ÿ'™' ! Q¡$¨¨^Ñ);¸aÑ)?Ñ"?Ó@Ñ@Ø˜.Ñ(ñ*ˆEð ˆDØ !ÐØ&×+Ñ+Ó-ˆLð $ dÒ*Ø&¬¯©°;°,¸uÑ2DÓ)EÑE�Ø(¨1¨^Ñ+;¸h¿l¹l»nÑ+LÑL�	Ü$'¨°Ñ):¸EÑ(AÓ$BÐ!Ø!�ð	 $ dÓ*ð
 ˆLr8   ÚhalflogisticrŸ  r(  )rE   rF   rG   r5   rö   r÷   r'  rˆ  r.   r/   r–  s             €r6   rC   zhalflogistic_gen.fit  s§   ø€ ð �8‰8�J Ô&Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ä8¸¸tØ9=¸tóEÑˆˆd�Fò	ôD —6‘6˜$“<ˆØÐØ˜$Šä" >¸ÄRÇVÁVÔLÐLØ‰Cð ˆCð !Ð,‘±*¸TÀ3Ó2Gˆà�EˆzÐr8   )rƒ   r„   r…   r†   rk   rr   rÞ   ru   r~   ry   r�   r  rò   rK   r   r   rC   rÐ  rÑ  s   @r6   r*  r*  ×  sV   ø„ ñò2ò'ò
9òòò ò<ò
?òð Ù˜MÓ*ó6ó +ó ô6r8   r*  rH  c                   ó€   ‡ — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Ze ee«      ˆ fd„«       «       Zˆ xZS )Úhalfnorm_genaF  A half-normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `halfnorm` is:

    .. math::

        f(x) = \sqrt{2/\pi} \exp(-x^2 / 2)

    for :math:`x >= 0`.

    `halfnorm` is a special case of `chi` with ``df=1``.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zhalfnorm_gen._shape_infon  r¦   r8   c                 ó8   — t        |j                  |¬«      «      S r4  r  rÖ   s      r6   rÙ   zhalfnorm_gen._rvsq  s   € Ü�<×/Ñ/°TÐ/Ó:Ó;Ð;r8   c                 óˆ   — t        j                  dt         j                  z  «      t        j                  | |z  dz  «      z  S r>  ©rP   rÿ   rñ   r·   r©   s     r6   rr   zhalfnorm_gen._pdft  s1   € ä�w‰w�sœ2Ÿ5™5‘yÓ!¤"§&¡&¨!¨¨A©¨c©Ó"2Ñ2Ð2r8   c                 óf   — dt        j                  dt         j                  z  «      z  ||z  dz  z
  S ©Nr”   r¶   rï   r©   s     r6   rÞ   zhalfnorm_gen._logpdfx  s+   € Ø”R—V‘V˜C¤§¡™IÓ&Ñ&¨¨1©¨S©Ñ0Ð0r8   c                 óX   — t        j                  |t        j                  d«      z  «      S r  ©rw   r   rP   rÿ   r©   s     r6   ru   zhalfnorm_gen._cdf{  s   € Ü�v‰v�aœ"Ÿ'™' !›*‘nÓ%Ð%r8   c                 ó$   — t        d|z   dz  «      S r  rÏ   r°   s     r6   r~   zhalfnorm_gen._ppf~  s   € Ü˜!˜A™#˜s™Ó#Ð#r8   c                 ó   — dt        |«      z  S r  rå   r©   s     r6   ry   zhalfnorm_gen._sf�  s   € Ø”8˜A“;‰Ðr8   c                 ó   — t        |dz  «      S r  rê   r  s     r6   r�   zhalfnorm_gen._isf„  s   € Ü˜˜1™‹~Ðr8   c                 óP  — t        j                  dt         j                  z  «      ddt         j                  z  z
  t        j                  d«      dt         j                  z
  z  t         j                  dz
  dz  z  dt         j                  dz
  z  t         j                  dz
  dz  z  fS )Nr¶   r   rU   r$  rÊ  r.  r†  ©rP   rÿ   rñ   rj   s    r6   r   zhalfnorm_gen._stats‡  sx   € Ü—‘˜œBŸE™E™	Ó"Ø�#”b—e‘e‘)‘Ü—‘˜“
˜AœbŸe™e™GÑ$¤b§e¡e¨A¡g°¡^Ñ3Ø”2—5‘5˜‘7‘œRŸU™U 1™W q™LÑ(ð*ð 	*r8   c                 óZ   — dt        j                  t         j                  dz  «      z  dz   S rP  rï   rj   s    r6   rò   zhalfnorm_gen._entropy�  s#   € Ø”2—6‘6œ"Ÿ%™% ™)Ó$Ñ$ SÑ(Ð(r8   c                 ó:  •— |j                  dd«      rt        ‰	| �  |g|¢­i |¤ŽS t        | |||«      \  }}}t	        j
                  |«      }|�$||k  rt        d|t        j                  ¬«      ‚|}n|}|�|}||fS t        j                  |d|¬«      dz  }||fS )Nr;  FÚhalfnormrŸ  rU   )ÚorderÚcenterr”   )
r3   rA   rC   rG  rP   r‡  rK  ri   rò  Úmoment)
rE   rF   rG   r5   rö   r÷   rˆ  r.   r/   r–  s
            €r6   rC   zhalfnorm_gen.fit�  sº   ø€ ð �8‰8�J Ô&Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ä8¸¸tØ9=¸tóEÑˆˆd�Fô —6‘6˜$“<ˆàÐØ˜$Šä" :°TÄÇÁÔHÐHØ‰CàˆCàÐØˆEð �EˆzÐô —L‘L ¨Q°sÔ;¸SÑ@ˆEà�EˆzÐr8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   r~   ry   r�   r   rò   rK   r   r   rC   rÐ  rÑ  s   @r6   rJ  rJ  X  s[   ø„ ñò*ó<ò3ò1ò&ò$òòò*ò)ð Ù˜MÓ*óó +ó ôr8   rJ  rZ  c                   ó@   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zy
)Úhypsecant_gena  A hyperbolic secant continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `hypsecant` is:

    .. math::

        f(x) = \frac{1}{\pi} \text{sech}(x)

    for a real number :math:`x`.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zhypsecant_gen._shape_infoÂ  r¦   r8   c                 óT   — dt         j                  t        j                  |«      z  z  S r  )rP   rñ   Úcoshr©   s     r6   rr   zhypsecant_gen._pdfÅ  s   € à”B—E‘Eœ"Ÿ'™' !›*Ñ$Ñ%Ð%r8   c                 óz   — dt         j                  z  t        j                  t        j                  |«      «      z  S r>  ©rP   rñ   rð  r·   r©   s     r6   ru   zhypsecant_gen._cdfÉ  s&   € Ø”2—5‘5‰yœŸ™¤2§6¡6¨!£9Ó-Ñ-Ð-r8   c                 óz   — t        j                  t        j                  t         j                  |z  dz  «      «      S r>  ©rP   rð   r  rñ   r°   s     r6   r~   zhypsecant_gen._ppfÌ  s&   € Ü�v‰v”b—f‘fœRŸU™U 1™W S™[Ó)Ó*Ð*r8   c                 ó|   — dt         j                  z  t        j                  t        j                  | «      «      z  S r>  rd  r©   s     r6   ry   zhypsecant_gen._sfÏ  s(   € Ø”2—5‘5‰yœŸ™¤2§6¡6¨1¨"£:Ó.Ñ.Ð.r8   c                 ó|   — t        j                  t        j                  t         j                  |z  dz  «      «       S r>  rf  r°   s     r6   r�   zhypsecant_gen._isfÒ  s)   € Ü—‘”r—v‘vœbŸe™e A™g c™kÓ*Ó+Ð+Ð+r8   c                 óR   — dt         j                  t         j                  z  dz  ddfS )Nr   r$  rU   r/  rj   s    r6   r   zhypsecant_gen._statsÕ  s!   € Ø”"—%‘%œŸ™‘+˜a‘-  AÐ%Ð%r8   c                 óN   — t        j                  dt         j                  z  «      S r  rï   rj   s    r6   rò   zhypsecant_gen._entropyØ  rž  r8   N)rƒ   r„   r…   r†   rk   rr   ru   r~   ry   r�   r   rò   r‡   r8   r6   r_  r_  ®  s/   „ ñò&ò&ò.ò+ò/ò,ò&ór8   r_  Ú	hypsecantc                   ó(   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zy)Úgausshyper_gena_  A Gauss hypergeometric continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `gausshyper` is:

    .. math::

        f(x, a, b, c, z) = C x^{a-1} (1-x)^{b-1} (1+zx)^{-c}

    for :math:`0 \le x \le 1`, :math:`a,b > 0`, :math:`c` a real number,
    :math:`z > -1`, and :math:`C = \frac{1}{B(a, b) F[2, 1](c, a; a+b; -z)}`.
    :math:`F[2, 1]` is the Gauss hypergeometric function
    `scipy.special.hyp2f1`.

    `gausshyper` takes :math:`a`, :math:`b`, :math:`c` and :math:`z` as shape
    parameters.

    %(after_notes)s

    References
    ----------
    .. [1] Armero, C., and M. J. Bayarri. "Prior Assessments for Prediction in
           Queues." *Journal of the Royal Statistical Society*. Series D (The
           Statistician) 43, no. 1 (1994): 139-53. doi:10.2307/2348939

    %(example)s

    c                 ó0   — |dkD  |dkD  z  ||k(  z  |dkD  z  S )Nr   r¿  r‡   )rE   r‹   rŒ   r  r*  s        r6   rc   zgausshyper_gen._argcheck   s'   € à�A‘˜!˜a™%Ñ  A¨¡FÑ+¨q°2©vÑ6Ð6r8   c                 ó  — t        dddt        j                  fd«      }t        dddt        j                  fd«      }t        ddt        j                   t        j                  fd«      }t        dddt        j                  fd«      }||||gS )	Nr‹   Fr   r
  rŒ   r  r*  r¿  rh   )rE   ri  rj  r!  Úizs        r6   rk   zgausshyper_gen._shape_info  sx   € Ü˜˜U Q¬¯© K°Ó@ˆÜ˜˜U Q¬¯© K°Ó@ˆÜ˜˜U¤b§f¡f W¬b¯f©fÐ$5°~ÓFˆÜ˜˜U R¬¯© L°.ÓAˆØ�B˜˜BÐÐr8   c                 ó´   — t        j                  ||«      t        j                  ||||z   | «      z  }d|z  ||dz
  z  z  d|z
  |dz
  z  z  d||z  z   |z  z  S r  ©rw   rn  Úhyp2f1)rE   rq   r‹   rŒ   r  r*  Únormalization_constants          r6   rr   zgausshyper_gen._pdf  sm   € Ü!#§¡¨¨A£´·±¸1¸aÀÀQÁÈÈÓ1KÑ!KÐØÐ)Ñ)¨A°°B±©KÑ7¸2À¹6ÀQÈÁWÑ:MÑMØ˜˜1™‘9˜q‘.ñ!ð 	"r8   c                 óî   — t        j                  ||z   |«      t        j                  ||«      z  }t        j                  |||z   ||z   |z   | «      }t        j                  ||||z   | «      }||z  |z  S rN   rr  )	rE   rb   r‹   rŒ   r  r*  r«  r@  rà  s	            r6   r  zgausshyper_gen._munp  sn   € Ü�g‰g�a˜‘c˜1‹o¤§¡¨¨1£Ñ-ˆÜ�i‰i˜˜1˜Q™3  !¡ A¡¨ rÓ*ˆÜ�i‰i˜˜1˜a ™c A 2Ó&ˆØ�3‰w˜‰}Ðr8   N)rƒ   r„   r…   r†   rc   rk   rr   r  r‡   r8   r6   rm  rm  ß  s   „ ñò@7ò ò"ó
r8   rm  Ú
gausshyperc                   ó`   — e Zd ZdZej
                  Zd„ Zd„ Zd„ Z	d„ Z
d„ Zd„ Zd„ Zdd	„Zd
„ Zy)Úinvgamma_gena_  An inverted gamma continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `invgamma` is:

    .. math::

        f(x, a) = \frac{x^{-a-1}}{\Gamma(a)} \exp(-\frac{1}{x})

    for :math:`x >= 0`, :math:`a > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    `invgamma` takes ``a`` as a shape parameter for :math:`a`.

    `invgamma` is a special case of `gengamma` with ``c=-1``, and it is a
    different parameterization of the scaled inverse chi-squared distribution.
    Specifically, if the scaled inverse chi-squared distribution is
    parameterized with degrees of freedom :math:`\nu` and scaling parameter
    :math:`\tau^2`, then it can be modeled using `invgamma` with
    ``a=`` :math:`\nu/2` and ``scale=`` :math:`\nu \tau^2/2`.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r	  rh   rj   s    r6   rk   zinvgamma_gen._shape_info:  r  r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  r  s      r6   rr   zinvgamma_gen._pdf=  r½  r8   c                 ór   — |dz    t        j                  |«      z  t        j                  |«      z
  d|z  z
  S r¿  ©rP   rð   rw   rÆ  r  s      r6   rÞ   zinvgamma_gen._logpdfA  s1   € Ø�1‘ˆvœŸ™˜q›	Ñ!¤B§J¡J¨q£MÑ1°C¸±EÑ9Ð9r8   c                 ó4   — t        j                  |d|z  «      S r  r¿  r  s      r6   ru   zinvgamma_gen._cdfD  s   € Ü�|‰|˜A˜s Q™wÓ'Ð'r8   c                 ó4   — dt        j                  ||«      z  S r  r,  r  s      r6   r~   zinvgamma_gen._ppfG  s   € Ø”R—_‘_ Q¨Ó*Ñ*Ð*r8   c                 ó4   — t        j                  |d|z  «      S r  r¼  r  s      r6   ry   zinvgamma_gen._sfJ  s   € Ü�{‰{˜1˜c A™gÓ&Ð&r8   c                 ó4   — dt        j                  ||«      z  S r  rë  r  s      r6   r�   zinvgamma_gen._isfM  s   € Ø”R—^‘^ A qÓ)Ñ)Ð)r8   c                 ó0  — t        |dkD  |fd„ t        j                  «      }t        |dkD  |fd„ t        j                  «      }d\  }}d|v r!t        |dkD  |fd„ t        j                  «      }d	|v r!t        |d
kD  |fd„ t        j                  «      }||||fS )Nr   c                 ó   — d| dz
  z  S r  r‡   r¹   s    r6   rç  z%invgamma_gen._stats.<locals>.<lambda>Q  s   € ¨r°Q¸±V©}€ r8   rU   c                 ó$   — d| dz
  dz  z  | dz
  z  S )Nr‰   rU   r¶   r‡   r¹   s    r6   rç  z%invgamma_gen._stats.<locals>.<lambda>R  s   € ¨r°Q¸±V¸a±KÑ/?À1ÀrÁ6Ñ/J€ r8   r  r  r†  c                 óD   — dt        j                  | dz
  «      z  | dz
  z  S )NrJ  r¶   rD  r‡  r¹   s    r6   rç  z%invgamma_gen._stats.<locals>.<lambda>Y  s    € ˜"œrŸw™w q¨2¡v›Ñ.°!°b±&Ñ9€ r8   r  r$  c                 ó0   — dd| z  dz
  z  | dz
  z  | dz
  z  S )Nr�  r	  g      &@rD  rJ  r‡   r¹   s    r6   rç  z%invgamma_gen._stats.<locals>.<lambda>]  s%   € ˜"  Q¡¨¡Ñ-°°R±Ñ8¸AÀ¹FÑC€ r8   rµ  )rE   r‹   r  r=  r>  rE  rF  s          r6   r   zinvgamma_gen._statsP  s¤   € Ü˜˜A™ ˜tÑ%<¼b¿f¹fÓEˆÜ˜˜A™ ˜tÑ%JÜŸ™ó ˆð ‰ˆˆBØ�'‰>ÜØ�A‘˜�tÙ9¼2¿6¹6óCˆBð �'‰>ÜØ�A‘˜�tÙCÄRÇVÁVóMˆBð �2�r˜2ˆ~Ðr8   c                 ó8   — d„ }d„ }t        |dk\  |f||¬«      }|S )Nc                 ón   — | | dz   t        j                  | «      z  z
  t        j                  | «      z   }|S r  r3  ©r‹   ró  s     r6   r¯  z&invgamma_gen._entropy.<locals>.regulara  s/   € Ø�Q˜‘W¤§¡ q£	Ñ)Ñ)¬B¯J©J°q«MÑ9ˆAØˆHr8   c                 óþ   — ddt        j                  | «      z  z
  t        j                  d«      z   t        j                  t         j                  «      z   dz  d| dz  z  z   | dz  dz  z   | dz  d	z  z
  | d
z  dz  z
  }|S )Nr   r†  rU   çUUUUUUå?r<  r³  rÀ  r´  r  rµ  r·  rï   rˆ  s     r6   rp  z)invgamma_gen._entropy.<locals>.asymptotice  s‚   € ð �aœŸ™˜q›	‘k‘/¤B§F¡F¨1£IÑ-´·±´r·u±u³Ñ=¸qÑ@Ø�q˜#‘v‘:ñØ ! 3¡ r¡	ñ*Ø,-¨s©F°2©Iñ6Ø89¸3¹¸s¹
ñCˆAàˆHr8   rq  rŒ  rì  )rE   r‹   r¯  rp  ró  s        r6   rò   zinvgamma_gen._entropy`  s)   € ò	ò	ô �q˜C‘x ! ¨¸Ô@ˆØˆr8   N©Úmvsk)rƒ   r„   r…   r†   r   r  r  rk   rr   rÞ   ru   r~   ry   r�   r   rò   r‡   r8   r6   rx  rx    sB   „ ñð: "×4Ñ4€MòEò*ò:ò(ò+ò'ò*óó r8   rx  Úinvgammac                   ó¢   ‡ — e Zd ZdZej
                  Zd„ Zdd„Zd„ Z	d„ Z
d„ Zd„ Zd„ Zd	„ Zˆ fd
„Zˆ fd„Zd„ Z ee«      ˆ fd„«       Zd„ Zˆ xZS )Úinvgauss_genaU  An inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `invgauss` is:

    .. math::

        f(x; \mu) = \frac{1}{\sqrt{2 \pi x^3}}
                    \exp\left(-\frac{(x-\mu)^2}{2 \mu^2 x}\right)

    for :math:`x \ge 0` and :math:`\mu > 0`.

    `invgauss` takes ``mu`` as a shape parameter for :math:`\mu`.

    %(after_notes)s

    A common shape-scale parameterization of the inverse Gaussian distribution
    has density

    .. math::

        f(x; \nu, \lambda) = \sqrt{\frac{\lambda}{2 \pi x^3}}
                    \exp\left( -\frac{\lambda(x-\nu)^2}{2 \nu^2 x}\right)

    Using ``nu`` for :math:`\nu` and ``lam`` for :math:`\lambda`, this
    parameterization is equivalent to the one above with ``mu = nu/lam``,
    ``loc = 0``, and ``scale = lam``.

    This distribution uses routines from the Boost Math C++ library for
    the computation of the ``ppf`` and ``isf`` methods. [1]_

    References
    ----------
    .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS ©NrC  Fr   r
  rh   rj   s    r6   rk   zinvgauss_gen._shape_infoŸ  r²  r8   c                 ó*   — |j                  |d|¬«      S ©Nr‰   r  ©Úwald©rE   rC  r×   rØ   s       r6   rÙ   zinvgauss_gen._rvs¢  s   € Ø× Ñ   S¨tÐ Ó4Ð4r8   c                 ó°   — dt        j                  dt         j                  z  |dz  z  «      z  t        j                  dd|z  z  ||z
  |z  dz  z  «      z  S )Nr‰   rU   rD  r<  rN  ©rE   rq   rC  s      r6   rr   zinvgauss_gen._pdf¥  sO   € ð ”2—7‘7˜1œRŸU™U™7 1 c¡6™>Ó*Ñ*¬2¯6©6°$¸¸!¹±*¸qÀ¹tÀR¹iÈ!¹^Ñ2KÓ+LÑLÐLr8   c                 óª   — dt        j                  dt         j                  z  «      z  dt        j                  |«      z  z
  ||z
  |z  dz  d|z  z  z
  S )Nr$  rU   rÊ  rï   r˜  s      r6   rÞ   zinvgauss_gen._logpdfª  sH   € Ø”B—F‘F˜1œRŸU™U™7“OÑ# c¬"¯&©&°«)¡mÑ3¸¸"¹¸b±yÀ1±nÀaÈÁcÑ6JÑJÐJr8   c                 óì   — dt        j                  |«      z  }t        |||z  dz
  z  «      }d|z  t        | ||z  dz   z  «      z   }|t        j                  t        j                  ||z
  «      «      z   S r1  )rP   rÿ   rÃ   r¦  r·   ©rE   rq   rC  r«  r‹   rŒ   s         r6   rã   zinvgauss_gen._logcdf±  sl   € Ø”"—'‘'˜!“*‰nˆÜ˜  R¡¨1¡Ñ-Ó.ˆØ�‰F”\ 3 $¨1¨r©6°Q©,Ñ"7Ó8Ñ8ˆØ”2—8‘8œBŸF™F 1 q¡5›MÓ*Ñ*Ð*r8   c                 óî   — dt        j                  |«      z  }t        |||z  dz
  z  «      }d|z  t        | ||z   z  |z  «      z   }|t        j                  t        j
                  ||z
  «       «      z   S r1  )rP   rÿ   rÍ   rÃ   r¦  r·   r›  s         r6   rç   zinvgauss_gen._logsf·  sn   € Ø”"—'‘'˜!“*‰nˆÜ˜  B¡¨!™|Ñ,Ó-ˆØ�‰F”\ 3 $¨!¨b©&¡/°BÑ"6Ó7Ñ7ˆØ”2—8‘8œRŸV™V A¨¡E›]˜NÓ+Ñ+Ð+r8   c                 óL   — t        j                  | j                  ||«      «      S rN   r6  r˜  s      r6   ry   zinvgauss_gen._sf½  s   € Ü�v‰v�d—k‘k ! RÓ(Ó)Ð)r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rO  r˜  s      r6   ru   zinvgauss_gen._cdfÀ  ó   € Ü�v‰v�d—l‘l 1 bÓ)Ó*Ð*r8   c                 ó–  •— t        j                  ddd¬«      5  t        j                  ||«      \  }}t        j                  t	        j
                  ||d«      «      }|dkD  }t	        j                  d||   z
  ||   d«      ||<   t        j                  |«      }t        ‰| �%  ||   ||   «      ||<   d d d «       |S # 1 sw Y   S xY w©Nr9  )r;  rr  r<  r   r”   )
rP   r<  rñ  rû   rn   Ú_invgauss_ppfÚ_invgauss_isfr¢  rA   r~   )rE   rq   rC  ÚppfÚi_wtÚi_nanr–  s         €r6   r~   zinvgauss_gen._ppfÃ  sº   ø€ Ü�[‰[ ¨xÀÔJñ 	;Ü×'Ñ'¨¨2Ó.‰EˆAˆrÜ—*‘*œS×.Ñ.¨q°"°aÓ8Ó9ˆCØ�s‘7ˆDÜ×)Ñ)¨!¨A¨d©G©)°R¸±X¸qÓAˆC�‰IÜ—H‘H˜S“MˆEÜ™™ a¨¡h°°5±	Ó:ˆC�‰J÷	;ð ˆ
÷	;ð ˆ
ús   šBB>Â>Cc                 óp  •— t        j                  ddd¬«      5  t        j                  ||«      \  }}t        j                  ||d«      }|dkD  }t        j
                  d||   z
  ||   d«      ||<   t        j                  |«      }t        ‰| �!  ||   ||   «      ||<   d d d «       |S # 1 sw Y   S xY wr¡  )	rP   r<  rñ  rn   r£  r¢  r¢  rA   r�   )rE   rq   rC  Úisfr¥  r¦  r–  s         €r6   r�   zinvgauss_gen._isfÍ  s±   ø€ Ü�[‰[ ¨xÀÔJñ 	;Ü×'Ñ'¨¨2Ó.‰EˆAˆrÜ×#Ñ# A r¨1Ó-ˆCØ�s‘7ˆDÜ×)Ñ)¨!¨A¨d©G©)°R¸±X¸qÓAˆC�‰IÜ—H‘H˜S“MˆEÜ™™ a¨¡h°°5±	Ó:ˆC�‰J÷	;ð ˆ
÷	;ð ˆ
ús   šBB+Â+B5c                 óF   — ||dz  dt        j                  |«      z  d|z  fS )NrD  r†  rÖ  r‡  )rE   rC  s     r6   r   zinvgauss_gen._stats×  s%   € Ø�2�s‘7˜AœbŸg™g b›k™M¨2¨b©5Ð0Ð0r8   c                 ó  •— |j                  dd«      }t        |t        «      s#t        | t        «      s|j	                  «       dk(  rt        ‰	| �  |g|¢­i |¤ŽS t        | |||«      \  }}}}	 |�|�t        ‰	| �  |g|¢­i |¤ŽS t        j                  ||z
  dk  «      rt        ddt        j                  ¬«      ‚||z
  }t        j                  |«      }|€*t        |«      t        j                  |dz  |dz  z
  «      z  }||z  }|||fS )Nr1   r;   r<   r   ÚinvgaussrŸ  r¿  )r=   r?   r*   Úwald_genr>   rA   rC   rG  rP   r£  rK  ri   rþ   r¤  r¥  )
rE   rF   rG   r5   r1   Úfshape_srö   r÷   Úfshape_nr–  s
            €r6   rC   zinvgauss_gen.fitÚ  s  ø€ à—‘˜( EÓ*ˆä�tœ\Ô*¬j¸¼xÔ.HØ—<‘<“> TÒ)Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ä'BÀ4ÈØCGÈó(OÑ$ˆˆh˜˜fð	ð ˆ<˜8Ð/Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3Ü�V‰V�D˜4‘K !‘OÔ$Ü˜z°¼"¿&¹&ÔAÐAà˜$‘;ˆDÜ—w‘w˜t“}ˆHØˆ~Ü˜T›¤b§f¡f¨T°R©Z¸(Àb¹.Ñ-HÓ&IÑJ�Ø &Ñ(ˆHØ˜˜vÐ%Ð%r8   c                 óê   — dt        j                  dt         j                  z  «      z   dt        j                  |«      z  z   }d|z  }t        j                  j                  |«      |z  }d|z  d|z  z
  S )zV
        Ref.: https://moser-isi.ethz.ch/docs/papers/smos-2012-10.pdf (eq. 9)
        r‰   rU   r†  r”   rÊ  )rP   rð   rñ   rw   rÐ  rÑ  )rE   rC  r‹   rõ  rŒ   s        r6   rò   zinvgauss_gen._entropyü  sg   € ð ”—‘˜œBŸE™E™	Ó"Ñ" Q¬¯©°«¡^Ñ3ˆð ˆb‰DˆÜ�J‰J×#Ñ# AÓ& qÑ(ˆØ�Q‰w˜˜q™Ñ Ð r8   r  )rƒ   r„   r…   r†   r   r  r  rk   rÙ   rr   rÞ   rã   rç   ry   ru   r~   r�   r   r   rC   rò   rÐ  rÑ  s   @r6   r�  r�  s  so   ø„ ñ(ðR "×4Ñ4€MòFó5òMò
Kò+ò,ò*ò+ôôò1ñ ˜MÓ*ó&ó +ð&öB!r8   r�  r«  c                   óN   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	dd	„Z
d
„ Zd„ Zd„ Zy)Úgeninvgauss_genab  A Generalized Inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `geninvgauss` is:

    .. math::

        f(x, p, b) = x^{p-1} \exp(-b (x + 1/x) / 2) / (2 K_p(b))

    where ``x > 0``, `p` is a real number and ``b > 0``\([1]_).
    :math:`K_p` is the modified Bessel function of second kind of order `p`
    (`scipy.special.kv`).

    %(after_notes)s

    The inverse Gaussian distribution `stats.invgauss(mu)` is a special case of
    `geninvgauss` with ``p = -1/2``, ``b = 1 / mu`` and ``scale = mu``.

    Generating random variates is challenging for this distribution. The
    implementation is based on [2]_.

    References
    ----------
    .. [1] O. Barndorff-Nielsen, P. Blaesild, C. Halgreen, "First hitting time
       models for the generalized inverse gaussian distribution",
       Stochastic Processes and their Applications 7, pp. 49--54, 1978.

    .. [2] W. Hoermann and J. Leydold, "Generating generalized inverse Gaussian
       random variates", Statistics and Computing, 24(4), p. 547--557, 2014.

    %(example)s

    c                 ó   — ||k(  |dkD  z  S r2  r‡   ©rE   rô  rŒ   s      r6   rc   zgeninvgauss_gen._argcheck2  s   € Ø�Q‘˜1˜q™5Ñ!Ð!r8   c                 ó    — t        ddt        j                   t        j                  fd«      }t        dddt        j                  fd«      }||gS )Nrô  Fr
  rŒ   r   rh   )rE   r…  rj  s      r6   rk   zgeninvgauss_gen._shape_info5  óB   € Ü˜˜U¤b§f¡f W¬b¯f©fÐ$5°~ÓFˆÜ˜˜U Q¬¯© K°Ó@ˆØ�Bˆxˆr8   c                 óî   — d„ }t        j                  |t         j                  g¬«      } ||||«      }t        j                  |«      j	                  «       rd}t        j                  |t        d¬«       |S )Nc                 ó0   — t        j                  | ||«      S rN   )r   Úgeninvgauss_logpdf©rq   rô  rŒ   s      r6   Úlogpdf_singlez.geninvgauss_gen._logpdf.<locals>.logpdf_single>  s   € Ü×,Ñ,¨Q°°1Ó5Ð5r8   r“  zjInfinite values encountered in scipy.special.kve(p, b). Values replaced by NaN to avoid incorrect results.r†  rE  )rP   r�  r–  r¢  r£  rH  rI  rJ  )rE   rq   rô  rŒ   rº  r*  rY   s          r6   rÞ   zgeninvgauss_gen._logpdf:  s\   € ò	6ô Ÿ™ ]¼B¿J¹J¸<ÔHˆá˜!˜Q Ó"ˆÜ�8‰8�A‹;�?‰?ÔðHˆCä�M‰M˜#œ~¸!Õ<Øˆr8   c                 óN   — t        j                  | j                  |||«      «      S rN   rÝ  ©rE   rq   rô  rŒ   s       r6   rr   zgeninvgauss_gen._pdfJ  rÞ  r8   c                 ó˜   ‡— | j                  ||«      \  Š}ˆfd„}t        j                  |t        j                  g¬«      } ||||«      S )Nc                 óô   •— t        j                  ||gt        «      j                  j	                  t        j
                  «      }t        j                  t        d|«      }t        j                  |‰| «      d   S )NÚ_geninvgauss_pdfr   )rP   r›  rR  rœ  r�  rž  r   rŸ  r   r   r¡  )rq   rô  rŒ   r¤  r¥  rÚ  s        €r6   Ú_cdf_singlez)geninvgauss_gen._cdf.<locals>._cdf_singleQ  s^   ø€ ÜŸ™ ! Q ¬Ó/×6Ñ6×>Ñ>¼v¿¹ÓOˆIÜ"×.Ñ.¬vÐ7IØ/8ó:ˆCô —>‘> # r¨1Ó-¨aÑ0Ð0r8   r“  )r–   rP   r�  r–  )rE   rq   rô  rŒ   rÙ  rÀ  rÚ  s         @r6   ru   zgeninvgauss_gen._cdfN  sE   ø€ Ø×"Ñ" 1 aÓ(‰ˆˆBô	1ô —l‘l ;¼¿
¹
°|ÔDˆá˜1˜a Ó#Ð#r8   c                 óJ   — t        |dkD  |||fd„ t        j                   «      S )Nr   c                 óV   — |dz
  t        j                  | «      z  || d| z  z   z  dz  z
  S r1  r2  r¹  s      r6   rç  z.geninvgauss_gen._logquasipdf.<locals>.<lambda>_  s,   € ¨1¨q©5´"·&±&¸³)Ñ*;¸aÀÀQÀqÁSÁ¹kÈ!¹mÑ*K€ r8   r  r¼  s       r6   Ú_logquasipdfzgeninvgauss_gen._logquasipdf\  s)   € ä˜!˜a™% ! Q¨ ÙKÜŸ6™6˜'ó#ð 	#r8   Nc                 óX  ‡	‡
— t        j                  |«      r+t        j                  |«      r| j                  ||||«      }�nR|j                  dk(  rA|j                  dk(  r2| j                  |j	                  «       |j	                  «       ||«      }�nt        j
                  ||«      \  }}t        |j                  |«      \  }Š	t        t        j                  |«      «      }t        j                  |«      }t        j                  ||gdgdgdgg¬«      Š
‰
j                  srt        ˆ	ˆ
fd„t        t        |«       d«      D «       «      }| j                  ‰
d   ‰
d   ||«      j!                  |«      ||<   ‰
j#                  «        ‰
j                  sŒr|dk(  r|j	                  «       }|S )Nr   Úmulti_indexÚreadonly©ÚflagsÚop_flagsc              3   ó\   •K  — | ]#  }‰|   s‰j                   |   n
t        d «      –— Œ% y ­wrN   ©rÅ  Úslice©r³  rÌ  ÚbcÚits     €€r6   rµ  z'geninvgauss_gen._rvs.<locals>.<genexpr>�  ó1   øè ø€ ò ;Ø !ð 79¸²e˜RŸ^™^¨AÒ.ÄÀtÃÓLñ ;ùó   ƒ),r   r‡   )rP   ró  Ú_rvs_scalarr×   rP  rñ  r   r·  r  rü  ÚemptyÚnditerÚfinishedÚtuplerý  r¤  rN  Úiternext)rE   rô  rŒ   r×   rØ   rö  ÚshpÚ
numsamplesÚidxrÎ  rÏ  s            @@r6   rÙ   zgeninvgauss_gen._rvsb  si  ù€ ô �;‰;�qŒ>œbŸk™k¨!œnØ×"Ñ" 1 a¨¨|Ó<ŠCØ�V‰V�qŠ[˜QŸV™V qš[Ø×"Ñ" 1§6¡6£8¨Q¯V©V«X°t¸\ÓJŠCô ×&Ñ& q¨!Ó,‰DˆAˆqô # 1§7¡7¨DÓ1‰GˆC�ô œRŸW™W S›\Ó*ˆJô —(‘(˜4“.ˆCä—‘˜A˜q˜6Ø"/ Ø&0 \°J°<Ð$@ôBˆBð —k’kô ô ;Ü%*¬C°«I¨:°qÓ%9ô;ó ;�à×+Ñ+¨B¨q©E°2°a±5¸*Ø,8ó:ß:A¹'À#»,ð �C‘à—‘”ð —k“kð  �2Š:Ø—(‘(“*ˆCØˆ
r8   c           	      óÔ  ‡ ‡‡‡3— d}|sd}‰dk  r‰ Šd}‰ j                  ‰‰«      }d}‰dk\  s‰dkD  rd}n0‰t        ddt        j                  d‰z
  «      z  dz  «      k\  rd}nd}t	        t        j
                  |«      «      }	t        j                  |	«      }
t        j                  |
«      }d}|�rÞ�rqd‰dz   z  ‰z  |z
  }d|z  ‰dz
  z  ‰z  dz
  }||dz  dz  z
  }d|dz  z  d	z  ||z  dz  z
  |z   }t        j                  | t        j                  d
|dz  z  «      z  dz  «      }t        j                  d|z  dz  «       }|t        j                  |dz  t        j                  dz  z   «      z  |dz  z
  }| t        j                  |dz  «      z  |dz  z
  }‰ j                  |‰‰«      Š3‰ j                  |‰‰«      ‰3z
  }‰ j                  |‰‰«      ‰3z
  }||z
  t        j                  d|z  «      z  }||z
  t        j                  d|z  «      z  }d}ˆˆ3ˆˆ fd„}|}nŠt        j                  d‰ j                  |‰‰«      z  «      }d‰z   t        j                  d‰z   dz  ‰dz  z   «      z   ‰z  }d}|t        j                  d‰ j                  |‰‰«      z  «      z  }d}ˆˆˆ fd„}||k\  rt        d«      ‚|dk  rt        d«      ‚d}||
k  �rr|
|z
  }||j                  |¬«      z  }|j                  |¬«      } |||z
  | z  z   } | |z  |z   }!dt        j                  |«      z   ||!«      k  }"t        j                   |"«      }#|#dkD  r|!|"   ||||#z    ||#z  }|dk(  r||
z  dk\  rd||
z  › d�}$t#        |$«      ‚|dz  }||
k  rŒ¶�nº‰d‰z
  z  }%t        j$                  |%d‰z  f«      }&t        j                  ‰ j                  |‰‰«      «      }'|'|%z  }(|%d‰z  k  rOt        j                  ‰ «      })‰dkD  r|)d‰z  ‰z  |%‰z  z
  z  ‰z  }*n$|)t        j                  d‰dz  z  «      z  }*nd\  })}*|&‰dz
  z  }+d|+z  t        j                  |& ‰z  dz  «      z  ‰z  },|(|*z   |,z   }-||
k  �rÖ|
|z
  }t        j                  |«      t        j                  |«      }!}.|j                  |¬«      }|-|j                  |¬«      z  } | |(k  }/t        j&                  |/«      | |(|*z   k  z  }0t        j&                  |/|0z  «      }1|%| |/   z  |(z  |!|/<   |'|.|/<   ‰dkD  r|%‰z  | |0   |(z
  ‰z  |)z  z   d‰z  z  |!|0<   n7‰t        j                  | |0   |(z
  t        j                  ‰«      z  «      z  |!|0<   |)|!|0   ‰dz
  z  z  |.|0<   t        j                  |& ‰z  dz  «      ‰| |1   |(z
  |*z
  z  d|+z  z  z
  }2d‰z  t        j                  |2«      z  |!|1<   |+t        j                  |!|1    ‰z  dz  «      z  |.|1<   t        j                  ||.z  «      ‰ j                  |!‰‰«      k  }"t!        |"«      }#|#dkD  r|!|"   ||||#z    ||#z  }||
k  r�ŒÖt        j(                  ||	«      }!|rd|!z  }!|!S )NFr   r   Tr”   rU   r†  r  é   iåÿÿÿrÕ  c                 ó0   •— ‰j                  | ‰‰«      ‰z
  S rN   ©rÃ  )rq   rŒ   Úlmrô  rE   s    €€€€r6   Úlogqpdfz,geninvgauss_gen._rvs_scalar.<locals>.logqpdfÒ  s   ø€ Ø×,Ñ,¨Q°°1Ó5¸Ñ:Ð:r8   c                 ó*   •— ‰j                  | ‰‰«      S rN   rÞ  )rq   rŒ   rô  rE   s    €€€r6   rà  z,geninvgauss_gen._rvs_scalar.<locals>.logqpdfà  s   ø€ Ø×,Ñ,¨Q°°1Ó5Ð5r8   zvmin must be smaller than vmax.zumax must be positive.r  iPÃ  z2Not a single random variate could be generated in zH attempts. Sampling does not appear to work for the provided parameters.)r   r   )Ú_moder‡  rP   rÿ   rÖ  Ú
atleast_1drü  ÚzerosÚarccosr!  rñ   rÃ  r·   rú   r  rð   r¥  r  r£  Úlogical_notrN  )4rE   rô  rŒ   rÙ  rØ   Ú
invert_resr?  Ú
ratio_unifÚ
mode_shiftÚsize1dÚNrq   Ú	simulatedÚa2Úa1Úp1Úq1Úphir\  Úroot1Úroot2Úd1Úd2ÚvminÚvmaxÚumaxrà  r  Úxplusrþ  r  r  rË  rÙ  ÚacceptÚ
num_acceptrY   r  ÚxsÚk1ÚA1Úk2ÚA2Úk3ÚA3rm  ró  Úcond1Úcond2Úcond3r*  rß  s4   ```                                                @r6   rÒ  zgeninvgauss_gen._rvs_scalar™  sö  û€ ð ˆ
ÙØˆJØˆqŠ5à�ˆAØˆJØ�J‰J�q˜!Óˆð ˆ
Ø�Š6�Q˜’Uà‰JØ”#�c˜1œrŸw™w q¨1¡u›~Ñ-°Ñ1Ó2Ò2à‰Jð ˆJô ”r—}‘} ZÓ0Ó1ˆÜ�G‰G�F‹OˆÜ�H‰H�Q‹KˆØˆ	ââØ˜1˜q™5‘\ AÑ%¨Ñ)�Ø˜‘U˜a !™e‘_ qÑ(¨1Ñ,�à˜"˜a™% !™)‘^�Ø˜˜Q™‘Y ‘^ b¨2¡g°¡kÑ1°AÑ5�Ü—i‘i  ¤b§g¡g¨c°B¸±E©kÓ&:Ñ :¸QÑ >Ó?�Ü—g‘g˜b 2™g¨™kÓ*Ð*�ØœRŸV™V C¨!¡G¬b¯e©e°a©iÑ$7Ó8Ñ8¸2À¹6ÑA�Ø˜œbŸf™f S¨1¡W›oÑ-°°Q±Ñ6�ð ×&Ñ& q¨!¨QÓ/�Ø×&Ñ& u¨a°Ó3°bÑ8�Ø×&Ñ& u¨a°Ó3°bÑ8�ð  ™	¤R§V¡V¨C°"©HÓ%5Ñ5�Ø ™	¤R§V¡V¨C°"©HÓ%5Ñ5�Ø�÷;ð ‘ô —v‘v˜c $×"3Ñ"3°A°q¸!Ó"<Ñ<Ó=�Ø˜a™%¤2§7¡7¨A°©E°A©:¸¸1¹Ñ+<Ó#=Ñ=¸qÑ@�Ø�àœrŸv™v c¨D×,=Ñ,=¸eÀQÈÓ,JÑ&JÓKÑK�Ø�ö6ð �tŠ|Ü Ð!BÓCÐCØ�qŠyÜ Ð!9Ó:Ð:àˆAØ˜a“-Ø˜	‘M�à˜<×/Ñ/°QÐ/Ó7Ñ7�Ø ×(Ñ(¨aÐ(Ó0�Ø˜D 4™K¨1Ñ,Ñ,�Ø˜!‘e˜a‘i�àœBŸF™F 1›I™+©°«Ñ5�ÜŸV™V F›^�
Ø ’>Ø<?À¹K�A�i ¨ZÑ!7Ð9Ø Ñ+�Ià ’N¨¨1©°ªð!Ø!" 1¡ ð &?ð?�Cô ' sÓ+Ð+Ø�Q‘�ð' ˜a•-ð, �a˜!‘e‘ˆBÜ—‘˜˜Q ™U˜Ó$ˆBÜ—‘˜×)Ñ)¨!¨Q°Ó2Ó3ˆBØ�b‘ˆBØ�A˜‘EŠzÜ—V‘V˜Q˜B“Z�Ø�q’5Ø  A¡¨™z¨B°©EÑ1Ñ2°QÑ6‘BàœbŸf™f Q¨¨A©¡XÓ.Ñ.‘Bà‘��BØ�a˜!‘e‘ˆBØ�R‘œ"Ÿ&™& "  q¡¨1¡Ó-Ñ-°Ñ1ˆBØ�R‘˜"‘ˆAð ˜a“-Ø˜	‘M�ÜŸ™ !›¤b§h¡h¨q£k�3�à ×(Ñ(¨aÐ(Ó0�Ø˜×,Ñ,°!Ð,Ó4Ñ4�Ø˜R™�ÜŸ™ uÓ-°°b¸2±g±Ñ>�ÜŸ™ u¨u¡}Ó5�à ! E¡(™]¨RÑ/��E‘
Ø��%‘à�q’5Ø"$ a¡%¨1¨U©8°b©=¸AÑ*=ÀÑ*BÑ"BÀaÈ!ÁeÑ!L�C˜’Jà!"¤R§V¡V¨Q¨u©X¸©]¼b¿f¹fÀQ»iÑ,GÓ%HÑ!H�C˜‘JØ  E¡
¨Q°©UÑ 3Ñ3��%‘ä—F‘F˜B˜3 ™7 Q™;Ó'¨!¨q°©x¸"©}¸rÑ/AÑ*BÀaÈ"ÁfÑ*MÑM�Ø !™V¤b§f¡f¨Q£iÑ/��E‘
Ø¤§¡¨¨E©
 {°Q¡¸Ñ':Ó ;Ñ;��%‘äŸ&™&  Q¡›-¨4×+<Ñ+<¸SÀ!ÀQÓ+GÑG�Ü  ›[�
Ø ’>Ø<?À¹K�A�i ¨ZÑ!7Ð9Ø Ñ+�Ið7 ˜a”-ô: �j‰j˜˜FÓ#ˆÙØ�c‘'ˆCØˆ
r8   c                 ó´   — |dk  r*|t        j                  |dz
  dz  |dz  z   «      dz   |z
  z  S t        j                  d|z
  dz  |dz  z   «      d|z
  z
  |z  S r1  r‡  r³  s      r6   râ  zgeninvgauss_gen._mode2  sf   € àˆqŠ5ØœŸ™  Q¡¨¡
¨Q°©TÑ 1Ó2°QÑ6¸Ñ:Ñ;Ð;ä—G‘G˜Q ™U Q™J¨¨A©Ñ-Ó.°!°a±%Ñ8¸AÑ=Ð=r8   c                 ó®  — t        j                  ||z   |«      }t        j                  ||«      }t        j                  |«      t        j                  |«      z  }|j	                  «       red}t        j                  |t        d¬«       t        j                  |t        j                  t        j                  ¬«      }||    ||    z  || <   |S ||z  }|S )Nz…Infinite values encountered in the moment calculation involving scipy.special.kve. Values replaced by NaN to avoid incorrect results.r†  rE  ©Údtype)rw   ÚkverP   rV   r£  rH  rI  rJ  Ú	full_liker  r–  )	rE   rb   rô  rŒ   r@  ÚdenomÚinf_valsrY   r?  s	            r6   r  zgeninvgauss_gen._munp9  s¯   € Ü�f‰f�Q˜‘U˜AÓˆÜ—‘�q˜!“ˆÜ—8‘8˜C“=¤2§8¡8¨E£?Ñ2ˆØ�<‰<Œ>ð.ˆCô �M‰M˜#œ~¸!Õ<Ü—‘˜S¤"§&¡&´·
±
Ô;ˆAØ ˜y™>¨E°8°)Ñ,<Ñ<ˆAˆxˆi‰Lð ˆð �e‘ˆAØˆr8   r  )rƒ   r„   r…   r†   rc   rk   rÞ   rr   ru   rÃ  rÙ   rÒ  râ  r  r‡   r8   r6   r±  r±    s=   „ ñ#òH"òò
ò -ò$ò#ó5ònWòr>ór8   r±  r®  c                   óf   ‡ — e Zd ZdZej
                  Zd„ Zd„ Zˆ fd„Z	d„ Z
d„ Zd„ Zd
d„Zd	„ Zˆ xZS )Únorminvgauss_gena  A Normal Inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `norminvgauss` is:

    .. math::

        f(x, a, b) = \frac{a \, K_1(a \sqrt{1 + x^2})}{\pi \sqrt{1 + x^2}} \,
                     \exp(\sqrt{a^2 - b^2} + b x)

    where :math:`x` is a real number, the parameter :math:`a` is the tail
    heaviness and :math:`b` is the asymmetry parameter satisfying
    :math:`a > 0` and :math:`|b| <= a`.
    :math:`K_1` is the modified Bessel function of second kind
    (`scipy.special.k1`).

    %(after_notes)s

    A normal inverse Gaussian random variable `Y` with parameters `a` and `b`
    can be expressed as a normal mean-variance mixture:
    ``Y = b * V + sqrt(V) * X`` where `X` is ``norm(0,1)`` and `V` is
    ``invgauss(mu=1/sqrt(a**2 - b**2))``. This representation is used
    to generate random variates.

    Another common parametrization of the distribution (see Equation 2.1 in
    [2]_) is given by the following expression of the pdf:

    .. math::

        g(x, \alpha, \beta, \delta, \mu) =
        \frac{\alpha\delta K_1\left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}
        {\pi \sqrt{\delta^2 + (x - \mu)^2}} \,
        e^{\delta \sqrt{\alpha^2 - \beta^2} + \beta (x - \mu)}

    In SciPy, this corresponds to
    `a = alpha * delta, b = beta * delta, loc = mu, scale=delta`.

    References
    ----------
    .. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions on
           Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
           pp. 151-157, 1978.

    .. [2] O. Barndorff-Nielsen, "Normal Inverse Gaussian Distributions and
           Stochastic Volatility Modelling", Scandinavian Journal of
           Statistics, Vol. 24, pp. 1-13, 1997.

    %(example)s

    c                 ó>   — |dkD  t        j                  |«      |k  z  S r2  )rP   Úabsolute©rE   r‹   rŒ   s      r6   rc   znorminvgauss_gen._argcheck„  s   € Ø�A‘œ"Ÿ+™+ a›.¨1Ñ,Ñ-Ð-r8   c                 ó    — t        dddt        j                  fd«      }t        ddt        j                   t        j                  fd«      }||gS rg  rh   rh  s      r6   rk   znorminvgauss_gen._shape_info‡  rW  r8   c                 ó&   •— t         ‰| �  |d¬«      S )N)r   r”   rM  rS  rT  s     €r6   r•  znorminvgauss_gen._fitstartŒ  s   ø€ ô ‰wÑ  ¨HÐ Ó5Ð5r8   c                 ó  — t        j                  |dz  |dz  z
  «      }|t         j                  z  }t        j                  d|«      }|t	        j
                  ||z  «      z  t        j                  ||z  ||z  z
  |z   «      z  |z  S r‹  )rP   rÿ   rñ   Úhypotrw   Úk1er·   )rE   rq   r‹   rŒ   rØ  Úfac1Úsqs          r6   rr   znorminvgauss_gen._pdf‘  ss   € Ü—‘˜˜1™˜q !™t™Ó$ˆØ”2—5‘5‰yˆÜ�X‰X�a˜‹^ˆØ”b—f‘f˜Q ™V“nÑ$¤r§v¡v¨a°©c°A°b±D©j¸5Ñ.@Ó'AÑAÀBÑFÐFr8   c           
      óÐ  — t        j                  |«      r6t        j                  | j                  |t         j
                  ||f¬«      d   S t        j                  |«      }t        j                  |«      }g }t        |||«      D ]K  \  }}}|j                  t        j                  | j                  |t         j
                  ||f¬«      d   «       ŒM t        j                  |«      S )NrM  r   )
rP   ró  r   r¡  rr   ri   rã  r¸  Úappendr›  )rE   rq   r‹   rŒ   Úresultr  Úa0r´  s           r6   ry   znorminvgauss_gen._sf—  s¹   € Ü�;‰;�qŒ>ä—>‘> $§)¡)¨Q´·±¸aÀ¸VÔDÀQÑGÐGä—‘˜aÓ ˆAÜ—‘˜aÓ ˆAØˆFÜ # A q¨!£ò @‘��R˜Ø—‘œiŸn™n¨T¯Y©Y¸¼B¿F¹FØ35°r°(ô<Ø<=ñ?õ @ð@ô —8‘8˜FÓ#Ð#r8   c                 óØ   ‡ — ˆ fd„}t        j                  |«      r
 ||||«      S g }t        |||«      D ]  \  }}}|j                   ||||«      «       Œ! t        j                  |«      S )Nc                 ót  •— ˆ
fd„}‰
j                  ||«      } ||||| «      }|dk(  r|S |dkD  r1d}|}||z   } ||||| «      dkD  rJd|z  }||z   } ||||| «      dkD  rŒn0d}|}||z
  } ||||| «      dk  rd|z  }||z
  } ||||| «      dk  rŒt        j                  |||||| f‰
j                  ¬«      }	|	S )Nc                 ó0   •— ‰j                  | ||«      |z
  S rN   ©ry   )rq   r‹   rŒ   r}   rE   s       €r6   Úeqz6norminvgauss_gen._isf.<locals>._isf_scalar.<locals>.eq§  s   ø€ à—x‘x  1 aÓ(¨1Ñ,Ð,r8   r   r   rU   )rG   r  )rþ   r   r<  r  )r}   r‹   rŒ   r"  ÚxmÚemÚdeltaÚleftÚrightr  rE   s             €r6   Ú_isf_scalarz*norminvgauss_gen._isf.<locals>._isf_scalar¥  s  ø€ ô-ð —‘˜1˜a“ˆBÙ�B˜˜1˜a“ˆBØ�QŠwà�	Ø�AŠvØ�Ø�Ø˜U™
�Ù˜  1 aÓ(¨1Ò,Ø˜e™G�EØ ™J�Eñ ˜  1 aÓ(¨1Ô,ð
 �Ø�Ø˜E‘z�Ù˜˜q ! QÓ'¨!Ò+Ø˜e™G�EØ ™:�Dñ ˜˜q ! QÓ'¨!Ó+ô —_‘_ R¨¨u¸A¸qÀ!¸9Ø*.¯)©)ô5ˆFàˆMr8   )rP   ró  r¸  r  r›  )	rE   r}   r‹   rŒ   r(  r  Úq0r  r´  s	   `        r6   r�   znorminvgauss_gen._isf¤  sk   ø€ ô	ôB �;‰;�qŒ>Ù˜q ! QÓ'Ð'àˆFÜ # A q¨!£ò 7‘��R˜Ø—‘™k¨"¨b°"Ó5Õ6ð7ä—8‘8˜FÓ#Ð#r8   c                 óÚ   — t        j                  |dz  |dz  z
  «      }t        j                  d|z  ||¬«      }||z  t        j                  |«      t        j                  ||¬«      z  z   S )NrU   r   )rC  r×   rØ   r×  )rP   rÿ   r«  rÙ  r  )rE   r‹   rŒ   r×   rØ   rØ  Úigs          r6   rÙ   znorminvgauss_gen._rvsÎ  so   € ô —‘˜˜1™˜q !™t™Ó$ˆÜ�\‰\˜Q˜u™W¨4¸lˆ\ÓKˆØ�2‰vœŸ™ ›¤d§h¡h°DØ<Hð '/ó 'Jñ Jñ Jð 	Jr8   c                 óÔ   — t        j                  |dz  |dz  z
  «      }||z  }|dz  |dz  z  }d|z  |t        j                  |«      z  z  }ddd|dz  z  |dz  z  z   z  |z  }||||fS )NrU   r†  rD  r   r$  r‡  )rE   r‹   rŒ   rØ  rþ   ÚvarianceÚskewnessÚkurtosiss           r6   r   znorminvgauss_gen._statsÖ  s„   € Ü—‘˜˜1™˜q !™t™Ó$ˆØ�5‰yˆØ�a‘4˜% ™(‘?ˆØ˜‘7˜a¤"§'¡'¨%£.Ñ0Ñ1ˆØ˜!˜a ! Q¡$™h¨¨A©™oÑ-Ñ.°Ñ6ˆØ�X˜x¨Ð1Ð1r8   r  )rƒ   r„   r…   r†   r   r  r  rc   rk   r•  rr   ry   r�   rÙ   r   rÐ  rÑ  s   @r6   r  r  L  sA   ø„ ñ4ðj "×4Ñ4€Mò.òô
6ò
Gò$ò($óTJö2r8   r  Únorminvgaussc                   ól   ‡ — e Zd ZdZej
                  Zd„ Zd„ Zd„ Z	d„ Z
d„ Zd„ Zd„ Zd	„ Zdˆ fd
„	Zˆ xZS )Úinvweibull_genuˆ  An inverted Weibull continuous random variable.

    This distribution is also known as the FrÃ©chet distribution or the
    type II extreme value distribution.

    %(before_notes)s

    Notes
    -----
    The probability density function for `invweibull` is:

    .. math::

        f(x, c) = c x^{-c-1} \exp(-x^{-c})

    for :math:`x > 0`, :math:`c > 0`.

    `invweibull` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    F.R.S. de Gusmao, E.M.M Ortega and G.M. Cordeiro, "The generalized inverse
    Weibull distribution", Stat. Papers, vol. 52, pp. 591-619, 2011.

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zinvweibull_gen._shape_info  r  r8   c                 ó    — t        j                  || dz
  «      }t        j                  || «      }t        j                  | «      }||z  |z  S r  ©rP   rÌ  r·   )rE   rq   r  Úxc1Úxc2s        r6   rr   zinvweibull_gen._pdf  sF   € ä�h‰h�q˜1˜"˜s™(Ó#ˆÜ�h‰h�q˜1˜"‹oˆÜ�f‰f�c�T‹lˆØ�3‰w˜‰}Ðr8   c                 ó\   — t        j                  || «      }t        j                  | «      S rN   r5  )rE   rq   r  r6  s       r6   ru   zinvweibull_gen._cdf  s#   € Ü�h‰h�q˜1˜"‹oˆÜ�v‰v�s�d‹|Ðr8   c                 ó8   — t        j                  || z   «       S rN   )rP   r  r
  s      r6   ry   zinvweibull_gen._sf  s   € Ü—‘˜!˜a˜R™%˜Ó Ð Ð r8   c                 ó\   — t        j                  t        j                  |«       d|z  «      S r?  )rP   rÌ  rð   r  s      r6   r~   zinvweibull_gen._ppf  s!   € Ü�x‰xœŸ™ ›˜
 D¨¡FÓ+Ð+r8   c                 ó<   — t        j                  | «       d|z  z  S ©Nr¿  r9  rÎ  s      r6   r�   zinvweibull_gen._isf  s   € Ü—‘˜1˜"“�  A¡Ñ&Ð&r8   c                 ó8   — t        j                  d||z  z
  «      S r^   rj  r|  s      r6   r  zinvweibull_gen._munp  s   € Ü�x‰x˜˜A ™E™	Ó"Ð"r8   c                 óT   — dt         z   t         |z  z   t        j                  |«      z
  S r^   r8  r~  s     r6   rò   zinvweibull_gen._entropy  s"   € Ø”‰xœ& 1™*Ñ$¤r§v¡v¨a£yÑ0Ð0r8   c                 ó2   •— |€dn|}t         ‰| �  ||¬«      S )N)r¶   rM  rS  )rE   rF   rG   r–  s      €r6   r•  zinvweibull_gen._fitstart  s#   ø€ à˜‰v¨4ˆÜ‰wÑ  ¨DÐ Ó1Ð1r8   rN   )rƒ   r„   r…   r†   r   r  r  rk   rr   ru   ry   r~   r�   r  rò   r•  rÐ  rÑ  s   @r6   r2  r2  â  sH   ø„ ñð: "×4Ñ4€MòEòòò!ò,ò'ò#ò1÷2ñ 2r8   r2  Ú
invweibullc                   ó<   — e Zd ZdZd„ Zd„ Zd
d„Zd„ Zd„ Zd„ Z	d	„ Z
y)Újf_skew_t_gena€  Jones and Faddy skew-t distribution.

    %(before_notes)s

    Notes
    -----
    The probability density function for `jf_skew_t` is:

    .. math::

        f(x; a, b) = C_{a,b}^{-1}
                    \left(1+\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{a+1/2}
                    \left(1-\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{b+1/2}

    for real numbers :math:`a>0` and :math:`b>0`, where
    :math:`C_{a,b} = 2^{a+b-1}B(a,b)(a+b)^{1/2}`, and :math:`B` denotes the
    beta function (`scipy.special.beta`).

    When :math:`a<b`, the distribution is negatively skewed, and when
    :math:`a>b`, the distribution is positively skewed. If :math:`a=b`, then
    we recover the `t` distribution with :math:`2a` degrees of freedom.

    `jf_skew_t` takes :math:`a` and :math:`b` as shape parameters.

    %(after_notes)s

    References
    ----------
    .. [1] M.C. Jones and M.J. Faddy. "A skew extension of the t distribution,
           with applications" *Journal of the Royal Statistical Society*.
           Series B (Statistical Methodology) 65, no. 1 (2003): 159-174.
           :doi:`10.1111/1467-9868.00378`

    %(example)s

    c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS rg  rh   rh  s      r6   rk   zjf_skew_t_gen._shape_infoM  rk  r8   c                 ó0  — d||z   dz
  z  t        j                  ||«      z  t        j                  ||z   «      z  }d|t        j                  ||z   |dz  z   «      z  z   |dz   z  }d|t        j                  ||z   |dz  z   «      z  z
  |dz   z  }||z  |z  S ©NrU   r   r”   )rw   rn  rP   rÿ   )rE   rq   r‹   rŒ   r  rô  rõ  s          r6   rr   zjf_skew_t_gen._pdfR  sœ   € Ø�!�a‘%˜!‘)ÑœrŸw™w q¨!›}Ñ,¬r¯w©w°q¸1±u«~Ñ=ˆØ�!”b—g‘g˜a !™e a¨1¡f™nÓ-Ñ-Ñ-°1°s±7Ñ;ˆØ�!”b—g‘g˜a !™e a¨1¡f™nÓ-Ñ-Ñ-°1°s±7Ñ;ˆØ�B‰w˜‰{Ðr8   Nc                 ó°   — |j                  |||«      }d|z  dz
  t        j                  ||z   «      z  }dt        j                  |d|z
  z  «      z  }||z  S r‹  )rn  rP   rÿ   )rE   r‹   rŒ   r×   rØ   rô  rõ  Úd3s           r6   rÙ   zjf_skew_t_gen._rvsX  sX   € Ø×Ñ˜q ! TÓ*ˆØ�"‰f�q‰jœBŸG™G A¨¡E›NÑ*ˆØ”—‘˜˜q 2™v™Ó'Ñ'ˆØ�B‰wˆr8   c                 ó~   — d|t        j                  ||z   |dz  z   «      z  z   dz  }t        j                  |||«      S ©Nr   rU   r”   )rP   rÿ   rw   r|  ©rE   rq   r‹   rŒ   r�  s        r6   ru   zjf_skew_t_gen._cdf^  s>   € Ø�”R—W‘W˜Q ™U Q¨!¡V™^Ó,Ñ,Ñ,°Ñ3ˆÜ�z‰z˜!˜Q Ó"Ð"r8   c                 ó~   — d|t        j                  ||z   |dz  z   «      z  z   dz  }t        j                  |||«      S rI  )rP   rÿ   rw   r~  rJ  s        r6   ry   zjf_skew_t_gen._sfb  s>   € Ø�”R—W‘W˜Q ™U Q¨!¡V™^Ó,Ñ,Ñ,°Ñ3ˆÜ�{‰{˜1˜a Ó#Ð#r8   c                 ó¸   — t         j                  |||«      }d|z  dz
  t        j                  ||z   «      z  }dt        j                  |d|z
  z  «      z  }||z  S r‹  )rn  r¤  rP   rÿ   )rE   r}   r‹   rŒ   rô  rõ  rG  s          r6   r~   zjf_skew_t_gen._ppff  sV   € Ü�X‰X�a˜˜AÓˆØ�"‰f�q‰jœBŸG™G A¨¡E›NÑ*ˆØ”—‘˜˜q 2™v™Ó'Ñ'ˆØ�B‰wˆr8   c                 ó¼   — d„ }|d|z  kD  |d|z  kD  z  |dk\  z  }t        ||||ft        j                  |t        j                  g¬«      t        j                  «      S )z­Returns the n-th moment(s) where all the following hold:

        - n >= 0
        - a > n / 2
        - b > n / 2

        The result is np.nan in all other cases.
        c                 ón  — ||z   d| z  z  }d| z  t        j                  ||«      z  }t        j                  | dz   «      }t        j                  |dz  dkD  dd«      }t        j                  |d| z  z   |z
  |d| z  z
  |z   «      }t        j
                  | |«      |z  |z  }||z  |j                  «       z  S )zgComputes E[T^(n_k)] where T is skew-t distributed with
            parameters a_k and b_k.
            r”   rU   r   r   r¿  )rw   rn  rP   rM  rO  r¹  r¥  )	Ún_kÚa_kÚb_kr@  r  Úindicesry  rÎ  Ú	sum_termss	            r6   Ú
nth_momentz'jf_skew_t_gen._munp.<locals>.nth_momentu  s´   € ð ˜‘9 #¨¡)Ñ,ˆCØ˜‘HœrŸw™w s¨CÓ0Ñ0ˆEä—i‘i  a¡Ó(ˆGÜ—(‘(˜7 Q™;¨™?¨B°Ó2ˆCÜ—‘˜˜c C™i™¨'Ñ1°3¸¸s¹±?ÀWÑ3LÓMˆAÜŸ™  WÓ-°Ñ3°aÑ7ˆIà˜‘; §¡£Ñ0Ð0r8   r”   r   r“  ©r   rP   r�  r–  r  )rE   rb   r‹   rŒ   rT  Únth_moment_valids         r6   r  zjf_skew_t_gen._munpl  sa   € ò	1ð   a¡™K¨A°°a±©KÑ8¸AÀ¹FÑCÐÜØØ��1ˆIÜ�L‰L˜¬R¯Z©Z¨LÔ9Ü�F‰Fó	
ð 	
r8   r  )rƒ   r„   r…   r†   rk   rr   rÙ   ru   ry   r~   r  r‡   r8   r6   rB  rB  (  s+   „ ñ#òHò
óò#ò$òó
r8   rB  Ú	jf_skew_tc                   óR   — e Zd ZdZej
                  Zd„ Zd„ Zd„ Z	d„ Z
d„ Zd„ Zd„ Zy	)
Újohnsonsb_gena!  A Johnson SB continuous random variable.

    %(before_notes)s

    See Also
    --------
    johnsonsu

    Notes
    -----
    The probability density function for `johnsonsb` is:

    .. math::

        f(x, a, b) = \frac{b}{x(1-x)}  \phi(a + b \log \frac{x}{1-x} )

    where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0`
    and :math:`x \in [0,1]`.  :math:`\phi` is the pdf of the normal
    distribution.

    `johnsonsb` takes :math:`a` and :math:`b` as shape parameters.

    %(after_notes)s

    %(example)s

    c                 ó   — |dkD  ||k(  z  S r2  r‡   r  s      r6   rc   zjohnsonsb_gen._argcheck­  rU  r8   c                 ó    — t        ddt        j                   t        j                  fd«      }t        dddt        j                  fd«      }||gS ©Nr‹   Fr
  rŒ   r   rh   rh  s      r6   rk   zjohnsonsb_gen._shape_info°  rµ  r8   c                 ól   — t        ||t        j                  |«      z  z   «      }|dz  |d|z
  z  z  |z  S r)  )rº   rw   r9  )rE   rq   r‹   rŒ   Útrms        r6   rr   zjohnsonsb_gen._pdfµ  s8   € ä˜˜AœbŸh™h q›k™MÑ)Ó*ˆØ�‰u�a˜˜1™‘g‰˜sÑ"Ð"r8   c                 óJ   — t        ||t        j                  |«      z  z   «      S rN   )rÀ   rw   r9  rt  s       r6   ru   zjohnsonsb_gen._cdfº  s   € Ü˜˜QœrŸx™x¨›{™]Ñ*Ó+Ð+r8   c                 óP   — t        j                  d|z  t        |«      |z
  z  «      S r  )rw   r5  rÇ   rƒ  s       r6   r~   zjohnsonsb_gen._ppf½  ó#   € Ü�x‰x˜˜a™¤9¨Q£<°!Ñ#3Ñ4Ó5Ð5r8   c                 óJ   — t        ||t        j                  |«      z  z   «      S rN   )rÊ   rw   r9  rt  s       r6   ry   zjohnsonsb_gen._sfÀ  s   € Ü˜˜AœbŸh™h q›k™MÑ)Ó*Ð*r8   c                 óP   — t        j                  d|z  t        |«      |z
  z  «      S r  )rw   r5  rÐ   rƒ  s       r6   r�   zjohnsonsb_gen._isfÃ  ra  r8   N)rƒ   r„   r…   r†   r   r  r  rc   rk   rr   ru   r~   ry   r�   r‡   r8   r6   rY  rY  �  s7   „ ñð6 "×4Ñ4€Mò"òò
#ò
,ò6ò+ó6r8   rY  Ú	johnsonsbc                   óB   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
dd	„Zy
)Újohnsonsu_gena-  A Johnson SU continuous random variable.

    %(before_notes)s

    See Also
    --------
    johnsonsb

    Notes
    -----
    The probability density function for `johnsonsu` is:

    .. math::

        f(x, a, b) = \frac{b}{\sqrt{x^2 + 1}}
                     \phi(a + b \log(x + \sqrt{x^2 + 1}))

    where :math:`x`, :math:`a`, and :math:`b` are real scalars; :math:`b > 0`.
    :math:`\phi` is the pdf of the normal distribution.

    `johnsonsu` takes :math:`a` and :math:`b` as shape parameters.

    The first four central moments are calculated according to the formulas
    in [1]_.

    %(after_notes)s

    References
    ----------
    .. [1] Taylor Enterprises. "Johnson Family of Distributions".
       https://variation.com/wp-content/distribution_analyzer_help/hs126.htm

    %(example)s

    c                 ó   — |dkD  ||k(  z  S r2  r‡   r  s      r6   rc   zjohnsonsu_gen._argcheckî  rU  r8   c                 ó    — t        ddt        j                   t        j                  fd«      }t        dddt        j                  fd«      }||gS r\  rh   rh  s      r6   rk   zjohnsonsu_gen._shape_infoñ  rµ  r8   c                 ó–   — ||z  }t        ||t        j                  |«      z  z   «      }|dz  t        j                  |dz   «      z  |z  S r  )rº   rP   Úarcsinhrÿ   )rE   rq   r‹   rŒ   rA  r^  s         r6   rr   zjohnsonsu_gen._pdfö  sI   € ð ˆq‰SˆÜ˜˜A¤§
¡
¨1£Ñ-Ñ-Ó.ˆØ�‰u”R—W‘W˜R ™V“_Ñ$ SÑ(Ð(r8   c                 óJ   — t        ||t        j                  |«      z  z   «      S rN   )rÀ   rP   rj  rt  s       r6   ru   zjohnsonsu_gen._cdfý  s   € Ü˜˜Q¤§¡¨A£Ñ.Ñ.Ó/Ð/r8   c                 óJ   — t        j                  t        |«      |z
  |z  «      S rN   )rP   ÚsinhrÇ   rƒ  s       r6   r~   zjohnsonsu_gen._ppf   ó   € Ü�w‰wœ	 !› qÑ(¨AÑ-Ó.Ð.r8   c                 óJ   — t        ||t        j                  |«      z  z   «      S rN   )rÊ   rP   rj  rt  s       r6   ry   zjohnsonsu_gen._sf  s   € Ü˜˜A¤§
¡
¨1£Ñ-Ñ-Ó.Ð.r8   c                 óJ   — t        j                  t        |«      |z
  |z  «      S rN   )rP   rm  rÐ   rt  s       r6   r�   zjohnsonsu_gen._isf  rn  r8   c                 óŒ  — d\  }}}}|dz  }t        j                  |«      }	||z  }
d|v r|	dz   t        j                  |
«      z  }d|v r7dt        j                  |«      z  |	t        j
                  d|
z  «      z  dz   z  }d|v rš|	dz  t        j                  |«      dz  z  }d	t        j                  |
«      z  }|	|	dz   z  t        j                  d	|
z  «      z  }t        j                  d«      d|	t        j
                  d|
z  «      z  z   d
z  z  }| ||z   z  |z  }d|v rœd	d|	z  z   }d|	dz  z  |	dz   z  t        j
                  d|
z  «      z  }|	dz  t        j
                  d|
z  «      z  }dd	|	dz  z  z   d|	d	z  z  z   |	dz  z   }dd|	t        j
                  d|
z  «      z  z   dz  z  }||z   ||z  z   |z  d	z
  }||||fS )N©NNNNr³  r?  r”   rË  rU   r   r  r†  rÊ  r  r…  r$  rÔ  )rP   r·   rm  rw   r  rb  rÿ   )rE   r‹   rŒ   r  rC  rD  rE  rF  Úbn2Úexpbn2Úa_brº  r»  r¼  r  rF  s                   r6   r   zjohnsonsu_gen._stats	  sá  € ð 1‰ˆˆC��Rà�‰fˆÜ—‘˜“ˆØ�!‰eˆà�'‰>Ø˜#‘+�¤§¡¨£Ñ,ˆBØ�'‰>Ø”b—h‘h˜s“mÑ# V¬B¯G©G°A°c±E«NÑ%:¸QÑ%>Ñ?ˆCØ�'‰>Ø˜‘œbŸh™h s›m¨SÑ0Ñ0ˆBØ”2—7‘7˜3“<‘ˆBØ˜6 A™:Ñ&¬¯©°°3±«Ñ7ˆBÜ—G‘G˜A“J ! f¬r¯w©w°q¸±u«~Ñ&=Ñ"=ÀÑ!EÑEˆEØ�˜˜R™‘ 5Ñ(ˆBØ�'‰>Ø�Q�v‘X‘ˆBØ�6˜1‘9‘ ¨¡
Ñ+¬b¯g©g°a¸±e«nÑ<ˆBØ˜‘œRŸW™W Q s¡U›^Ñ+ˆBØ�a˜ ™	‘kÑ! A f¨a¡i¡KÑ/°&¸!±)Ñ;ˆBØ�q˜6¤"§'¡'¨!¨C©%£.Ñ0Ñ0°1Ñ4Ñ4ˆEØ�r‘'˜B˜r™E‘/ UÑ*¨QÑ.ˆBØ�3˜˜BˆÐr8   Nr  )rƒ   r„   r…   r†   rc   rk   rr   ru   r~   ry   r�   r   r‡   r8   r6   rf  rf  Ê  s0   „ ñ"òF"òò
)ò0ò/ò/ò/ôr8   rf  Ú	johnsonsuc                   óV   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zdd„Zdd„Zy)Ú
landau_gena4  A Landau continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `landau` ([1]_, [2]_) is:

    .. math::

        f(x) = \frac{1}{\pi}\int_0^\infty \exp(-t \log t - xt)\sin(\pi t) dt

    for a real number :math:`x`.

    %(after_notes)s

    Often (e.g. [2]_), the Landau distribution is parameterized in terms of a
    location parameter :math:`\mu` and scale parameter :math:`c`, the latter of
    which *also* introduces a location shift. If ``mu`` and ``c`` are used to
    represent these parameters, this corresponds with SciPy's parameterization
    with ``loc = mu + 2*c / np.pi * np.log(c)`` and ``scale = c``.

    This distribution uses routines from the Boost Math C++ library for
    the computation of the ``pdf``, ``cdf``, ``ppf``, ``sf`` and ``isf``
    methods. [1]_

    References
    ----------
    .. [1] Landau, L. (1944). "On the energy loss of fast particles by
           ionization". J. Phys. (USSR). 8: 201.
    .. [2] "Landau Distribution", Wikipedia,
           https://en.wikipedia.org/wiki/Landau_distribution
    .. [3] Chambers, J. M., Mallows, C. L., & Stuck, B. (1976).
           "A method for simulating stable random variables."
           Journal of the American Statistical Association, 71(354), 340-344.
    .. [4] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.
    .. [5] Yoshimura, T. "Numerical Evaluation and High Precision Approximation
           Formula for Landau Distribution".
           :doi:`10.36227/techrxiv.171822215.53612870/v2`

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zlandau_gen._shape_infoV  r¦   r8   c                  ó   — y)NgÚïXÐ(û@r‡   rj   s    r6   rò   zlandau_gen._entropyY  s   € à"r8   c                 ó0   — t        j                  |dd«      S r”  )rn   Ú_landau_pdfr©   s     r6   rr   zlandau_gen._pdf]  r–  r8   c                 ó0   — t        j                  |dd«      S r”  )rn   Ú_landau_cdfr©   s     r6   ru   zlandau_gen._cdf`  r–  r8   c                 ó0   — t        j                  |dd«      S r”  )rn   Ú
_landau_sfr©   s     r6   ry   zlandau_gen._sfc  s   € Ü�~‰~˜a  AÓ&Ð&r8   c                 ó0   — t        j                  |dd«      S r”  )rn   Ú_landau_ppfr  s     r6   r~   zlandau_gen._ppff  r–  r8   c                 ó0   — t        j                  |dd«      S r”  )rn   Ú_landau_isfr  s     r6   r�   zlandau_gen._isfi  r–  r8   c                 ó~   — t         j                  t         j                  t         j                  t         j                  fS rN   r›  rj   s    r6   r   zlandau_gen._statsl  rœ  r8   c                 ó0   — |dkD  rt         j                  S dS r”  r›  ra   s     r6   r  zlandau_gen._munpo  s   € Ø˜QšŒr�v‰vÐ% AÐ%r8   Nc                 óŽ   — t        |t        «      r|j                  «       }t        j                  |g d¢«      \  }}}|||z
  dz  fS r   r¥  r§  s         r6   r•  zlandau_gen._fitstartr  r«  r8   c                 ó†  — t         j                  dz  }|j                  t         j                   dz  t         j                  dz  |¬«      }|j                  |¬«      }dt         j                  z  ||z   t        j                  |«      z  t        j
                  ||z  t        j                  |«      z  ||z   z  «      z
  z  }|S )NrU   r  )rP   rñ   r  r6  r  rð   r!  )rE   r×   rØ   Úpi_2ÚUÚWÚSs          r6   rÙ   zlandau_gen._rvsy  s¢   € ä�u‰u�q‰yˆØ× Ñ ¤"§%¡% ¨!¡¬R¯U©U°Q©Y¸TÐ ÓBˆØ×-Ñ-°4Ð-Ó8ˆØ”—‘‰I˜$ ™(¤b§f¡f¨Q£iÑ/ÜŸ6™6 4¨!¡8¬b¯f©f°Q«iÑ#7¸DÀ1¹HÑ"EÓFñGñ Hˆàˆr8   rN   r  )rƒ   r„   r…   r†   rk   rò   rr   ru   ry   r~   r�   r   r  r•  rÙ   r‡   r8   r6   rx  rx  *  s?   „ ñ*òVò#ò(ò(ò'ò(ò(ò.ò&ó"ôr8   rx  Úlandauc                   ór   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Ze eed¬«      d„ «       «       Zy)Úlaplace_gena
  A Laplace continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `laplace` is

    .. math::

        f(x) = \frac{1}{2} \exp(-|x|)

    for a real number :math:`x`.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zlaplace_gen._shape_infoš  r¦   r8   Nc                 ó*   — |j                  dd|¬«      S )Nr   r   r  )ÚlaplacerÖ   s      r6   rÙ   zlaplace_gen._rvs�  s   € Ø×#Ñ# A q¨tÐ#Ó4Ð4r8   c                 óF   — dt        j                  t        |«       «      z  S r  )rP   r·   rŽ  r©   s     r6   rr   zlaplace_gen._pdf   s   € à”2—6‘6œ3˜q›6˜'“?Ñ"Ð"r8   c           	      óî   — t        j                  d¬«      5  t        j                  |dkD  ddt        j                  | «      z  z
  dt        j                  |«      z  «      cd d d «       S # 1 sw Y   y xY w)Nr9  rq  r   r‰   r”   )rP   r<  rO  r·   r©   s     r6   ru   zlaplace_gen._cdf¤  sZ   € Ü�[‰[˜hÔ'ñ 	HÜ—8‘8˜A ™E 3¨¬R¯V©V°Q°B«Z©Ñ#7¸¼R¿V¹VÀA»Y¹ÓG÷	H÷ 	Hò 	Hús   —A
A+Á+A4c                 ó&   — | j                  | «      S rN   ©ru   r©   s     r6   ry   zlaplace_gen._sf¨  s   € à�y‰y˜!˜‹}Ðr8   c                 ó–   — t        j                  |dkD  t        j                  dd|z
  z  «       t        j                  d|z  «      «      S rî   ©rP   rO  rð   r°   s     r6   r~   zlaplace_gen._ppf¬  s8   € Ü�x‰x˜˜C™¤"§&¡&¨¨A¨a©C©£/Ð!1´2·6±6¸!¸A¹#³;Ó?Ð?r8   c                 ó&   — | j                  |«       S rN   rŒ  r°   s     r6   r�   zlaplace_gen._isf¯  s   € à—	‘	˜!“ˆ}Ðr8   c                  ó   — y)N)r   rU   r   r†  r‡   rj   s    r6   r   zlaplace_gen._stats³  s   € Ør8   c                 ó2   — t        j                  d«      dz   S r‹  r2  rj   s    r6   rò   zlaplace_gen._entropy¶  s   € Ü�v‰v�a‹y˜‰{Ðr8   zÒ        This function uses explicit formulas for the maximum likelihood
        estimation of the Laplace distribution parameters, so the keyword
        arguments `loc`, `scale`, and `optimizer` are ignored.

ró   c                 óÎ   — t        | |||«      \  }}}|€t        j                  |«      }|€7t        j                  t        j                  ||z
  «      «      t        |«      z  }||fS rN   )rG  rP   Úmedianr¥  rŽ  r¤  )rE   rF   rG   r5   rö   r÷   s         r6   rC   zlaplace_gen.fit¹  sf   € ô 9¸¸tØ9=¸tóEÑˆˆd�Fð ˆ<Ü—9‘9˜T“?ˆDàˆ>Ü—f‘fœRŸV™V D¨4¡KÓ0Ó1´S¸³YÑ>ˆFà�Vˆ|Ðr8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   ru   ry   r~   r�   r   rò   rK   r
   r   rC   r‡   r8   r6   r�  r�  †  sa   „ ñò&ó5ò#òHòò@òòòð Ù ð 6Fô Gñó	Gó ñ
r8   r�  r’  c                   óF   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zy)Úlaplace_asymmetric_genuñ  An asymmetric Laplace continuous random variable.

    %(before_notes)s

    See Also
    --------
    laplace : Laplace distribution

    Notes
    -----
    The probability density function for `laplace_asymmetric` is

    .. math::

       f(x, \kappa) &= \frac{1}{\kappa+\kappa^{-1}}\exp(-x\kappa),\quad x\ge0\\
                    &= \frac{1}{\kappa+\kappa^{-1}}\exp(x/\kappa),\quad x<0\\

    for :math:`-\infty < x < \infty`, :math:`\kappa > 0`.

    `laplace_asymmetric` takes ``kappa`` as a shape parameter for
    :math:`\kappa`. For :math:`\kappa = 1`, it is identical to a
    Laplace distribution.

    %(after_notes)s

    Note that the scale parameter of some references is the reciprocal of
    SciPy's ``scale``. For example, :math:`\lambda = 1/2` in the
    parameterization of [1]_ is equivalent to ``scale = 2`` with
    `laplace_asymmetric`.

    References
    ----------
    .. [1] "Asymmetric Laplace distribution", Wikipedia
            https://en.wikipedia.org/wiki/Asymmetric_Laplace_distribution

    .. [2] Kozubowski TJ and PodgÃ³rski K. A Multivariate and
           Asymmetric Generalization of Laplace Distribution,
           Computational Statistics 15, 531--540 (2000).
           :doi:`10.1007/PL00022717`

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS )NÚkappaFr   r
  rh   rj   s    r6   rk   z"laplace_asymmetric_gen._shape_infoý  s   € Ü˜7 E¨A¬r¯v©v¨;¸ÓGÐHÐHr8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  ©rE   rq   r¡  s      r6   rr   zlaplace_asymmetric_gen._pdf   s   € Ü�v‰v�d—l‘l 1 eÓ,Ó-Ð-r8   c                 ó‚   — d|z  }|t        j                  |dk\  | |«      z  }|t        j                  ||z   «      z  }|S r�  r˜  )rE   rq   r¡  Úkapinvrz  s        r6   rÞ   zlaplace_asymmetric_gen._logpdf  sD   € Ø�5‘ˆØ”"—(‘(˜1 ™6 E 6¨6Ó2Ñ2ˆØŒr�v‰v�e˜F‘lÓ#Ñ#ˆØˆ
r8   c                 óÂ   — d|z  }||z   }t        j                  |dk\  dt        j                  | |z  «      ||z  z  z
  t        j                  ||z  «      ||z  z  «      S r�  ©rP   rO  r·   ©rE   rq   r¡  r¥  Ú
kappkapinvs        r6   ru   zlaplace_asymmetric_gen._cdf	  sf   € Ø�5‘ˆØ˜6‘\ˆ
Ü�x‰x˜˜Q™ØœBŸF™F A 2 e¡8Ó,¨f°ZÑ.?Ñ@Ñ@ÜŸ™˜q ™xÓ(¨%°
Ñ*:Ñ;ó=ð 	=r8   c           	      óÂ   — d|z  }||z   }t        j                  |dk\  t        j                  | |z  «      ||z  z  dt        j                  ||z  «      ||z  z  z
  «      S r�  r§  r¨  s        r6   ry   zlaplace_asymmetric_gen._sf  sh   € Ø�5‘ˆØ˜6‘\ˆ
Ü�x‰x˜˜Q™ÜŸ™ ˜r %™xÓ(¨&°Ñ*;Ñ<ØœBŸF™F 1 V¡8Ó,¨e°JÑ.>Ñ?Ñ?óAð 	Ar8   c                 óÈ   — d|z  }||z   }t        j                  |||z  k\  t        j                  d|z
  |z  |z  «       |z  t        j                  ||z  |z  «      |z  «      S r^   r˜  ©rE   r}   r¡  r¥  r©  s        r6   r~   zlaplace_asymmetric_gen._ppf  sm   € Ø�5‘ˆØ˜6‘\ˆ
Ü�x‰x˜˜U :Ñ-Ñ-ÜŸ™  Q¡¨
Ñ 2°5Ñ 8Ó9Ð9¸&Ñ@ÜŸ™˜q ™|¨EÑ1Ó2°5Ñ8ó:ð 	:r8   c                 óÈ   — d|z  }||z   }t        j                  |||z  k  t        j                  ||z  |z  «       |z  t        j                  d|z
  |z  |z  «      |z  «      S r^   r˜  r¬  s        r6   r�   zlaplace_asymmetric_gen._isf  so   € Ø�5‘ˆØ˜6‘\ˆ
Ü�x‰x˜˜V JÑ.Ñ.ÜŸ™  *¡¨UÑ 2Ó3Ð3°FÑ:ÜŸ™  A¡ zÑ1°%Ñ7Ó8¸Ñ>ó@ð 	@r8   c                 ó`  — d|z  }||z
  }||z  ||z  z   }ddt        j                  |d«      z
  z  t        j                  dt        j                  |d«      z   d«      z  }ddt        j                  |d«      z   z  t        j                  dt        j                  |d«      z   d«      z  }||||fS )	Nr   r¶   r…  r$  rÊ  r�  r.  rU   rÞ  )rE   r¡  r¥  Úmnr§  rE  rF  s          r6   r   zlaplace_asymmetric_gen._stats%  s¨   € Ø�5‘ˆØ�e‰^ˆØ�V‰m˜e E™kÑ)ˆØ�!”B—H‘H˜U AÓ&Ñ&Ñ'¬¯©°´2·8±8¸EÀ1Ó3EÑ1EÀsÓ(KÑKˆØ�!”B—H‘H˜U AÓ&Ñ&Ñ'¬¯©°´2·8±8¸EÀ1Ó3EÑ1EÀqÓ(IÑIˆØ�3˜˜BˆÐr8   c                 ó>   — dt        j                  |d|z  z   «      z   S r^   r2  ©rE   r¡  s     r6   rò   zlaplace_asymmetric_gen._entropy-  s   € Ø”2—6‘6˜%  %¡™-Ó(Ñ(Ð(r8   Nr  r‡   r8   r6   rŸ  rŸ  Ñ  s8   „ ñ*òVIò.òò=òAò:ò@òó)r8   rŸ  Úlaplace_asymmetricc                 ó¦  — t        |t        «      st        j                  |«      }|j	                  dd «      }|j	                  dd «      }| j
                  r$t        | j
                  j                  d«      «      nd}g }g }| j
                  rŒ| j
                  j                  dd«      j                  «       }	t        |	«      D ]T  \  }
}dt        |
«      z   }|d|z   d|z   g}t        ||«      }|j                  |«       |j                  |«       |€ŒP|||<   ŒV dd	d
dddh|£}t        |«      j                  |«      }|rt        d|› d�«      ‚t        |«      |kD  rt        d«      ‚d ||h|£vrt!        d«      ‚t        |t        «      r|j#                  «       n|}t        j$                  |«      j'                  «       st)        d«      ‚|g|¢|‘|‘­S )Nrö   r÷   ú,r   ú r�  Úfix_r.   r/   r0   r1   zUnknown keyword arguments: r2   zToo many positional arguments.rø   rù   )r?   r*   rP   rû   r=   Úshapesr¤  ÚsplitrV  Ú	enumerateÚstrr   r  ÚsetÚ
differencer4   r  r“  rü   rý   rú   )ÚdistrF   rG   r5   rö   r÷   Ú
num_shapesÚfshape_keysÚfshapesr·  rÌ  r  ÚkeyÚnamesrx  Ú
known_keysÚunknown_keysÚ
uncensoreds                     r6   rG  rG  4  sÕ  € Ü�dœLÔ)Ü�z‰z˜$Óˆà�8‰8�F˜DÓ!€DØ�X‰X�h Ó%€Fà04·²”�T—[‘[×&Ñ& sÓ+Ô,À€JØ€KØ€Gð
 ‡{‚{Ø—‘×$Ñ$ S¨#Ó.×4Ñ4Ó6ˆÜ˜fÓ%ò 	 ‰DˆAˆqØœ˜A›‘,ˆCØ˜# ™' 6¨A¡:Ð.ˆEÜ& t¨UÓ3ˆCØ×Ñ˜sÔ#Ø�N‰N˜3ÔØ‰Ø��S’	ð	 ð ˜ +¨xØ˜(ð2Ø%0ð2€Jä�t“9×'Ñ'¨
Ó3€LÙÜÐ5°l°^À1ÐEÓFÐFä
ˆ4ƒy�:ÒÜÐ8Ó9Ð9à�D˜&Ð+ 7Ð+Ñ+ô ð 'ó (ð 	(ô &0°´lÔ%C�—‘Ô!È€JÜ�;‰;�zÓ"×&Ñ&Ô(ÜÐ?Ó@Ð@àÐ)�7Ð)˜DÐ) &Ñ)Ð)r8   c                   óR   — e Zd ZdZej
                  Zd„ Zd„ Zd„ Z	d„ Z
d„ Zd„ Zd„ Zy	)
Úlevy_genag  A Levy continuous random variable.

    %(before_notes)s

    See Also
    --------
    levy_stable, levy_l

    Notes
    -----
    The probability density function for `levy` is:

    .. math::

        f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp\left(-\frac{1}{2x}\right)

    for :math:`x > 0`.

    This is the same as the Levy-stable distribution with :math:`a=1/2` and
    :math:`b=1`.

    %(after_notes)s

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import levy
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)

    Calculate the first four moments:

    >>> mean, var, skew, kurt = levy.stats(moments='mvsk')

    Display the probability density function (``pdf``):

    >>> # `levy` is very heavy-tailed.
    >>> # To show a nice plot, let's cut off the upper 40 percent.
    >>> a, b = levy.ppf(0), levy.ppf(0.6)
    >>> x = np.linspace(a, b, 100)
    >>> ax.plot(x, levy.pdf(x),
    ...        'r-', lw=5, alpha=0.6, label='levy pdf')

    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.

    Freeze the distribution and display the frozen ``pdf``:

    >>> rv = levy()
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

    Check accuracy of ``cdf`` and ``ppf``:

    >>> vals = levy.ppf([0.001, 0.5, 0.999])
    >>> np.allclose([0.001, 0.5, 0.999], levy.cdf(vals))
    True

    Generate random numbers:

    >>> r = levy.rvs(size=1000)

    And compare the histogram:

    >>> # manual binning to ignore the tail
    >>> bins = np.concatenate((np.linspace(a, b, 20), [np.max(r)]))
    >>> ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.set_xlim([x[0], x[-1]])
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zlevy_gen._shape_info°  r¦   r8   c                 ó˜   — dt        j                  dt         j                  z  |z  «      z  |z  t        j                  dd|z  z  «      z  S ©Nr   rU   r¿  rN  r©   s     r6   rr   zlevy_gen._pdf³  s=   € à”2—7‘7˜1œRŸU™U™7 1™9Ó%Ñ%¨Ñ)¬B¯F©F°2°q¸±s±8Ó,<Ñ<Ð<r8   c                 óX   — t        j                  t        j                  d|z  «      «      S r  )rw   ÚerfcrP   rÿ   r©   s     r6   ru   zlevy_gen._cdf·  s   € ä�w‰w”r—w‘w˜s Q™wÓ'Ó(Ð(r8   c                 óX   — t        j                  t        j                  d|z  «      «      S r  rR  r©   s     r6   ry   zlevy_gen._sf»  s   € Ü�v‰v”b—g‘g˜c A™gÓ&Ó'Ð'r8   c                 ó.   — t        |dz  «      }d||z  z  S ©NrU   r‰   rê   ©rE   r}   rx  s      r6   r~   zlevy_gen._ppf¾  s   € ä˜˜!™‹nˆØ�c˜C‘iÑ Ð r8   c                 ó>   — ddt        j                  |«      dz  z  z  S r1  )rw   Úerfinvr  s     r6   r�   zlevy_gen._isfÃ  s   € Ø�!”B—I‘I˜a“L !‘OÑ#Ñ$Ð$r8   c                 ó~   — t         j                  t         j                  t         j                  t         j                  fS rN   r  rj   s    r6   r   zlevy_gen._statsÆ  rœ  r8   N©rƒ   r„   r…   r†   r   r  r  rk   rr   ru   ry   r~   r�   r   r‡   r8   r6   rÇ  rÇ  e  s9   „ ñGðP "×4Ñ4€Mòò=ò)ò(ò!ò
%ó.r8   rÇ  Úlevyc                   óR   — e Zd ZdZej
                  Zd„ Zd„ Zd„ Z	d„ Z
d„ Zd„ Zd„ Zy	)
Ú
levy_l_gena“  A left-skewed Levy continuous random variable.

    %(before_notes)s

    See Also
    --------
    levy, levy_stable

    Notes
    -----
    The probability density function for `levy_l` is:

    .. math::
        f(x) = \frac{1}{|x| \sqrt{2\pi |x|}} \exp{ \left(-\frac{1}{2|x|} \right)}

    for :math:`x < 0`.

    This is the same as the Levy-stable distribution with :math:`a=1/2` and
    :math:`b=-1`.

    %(after_notes)s

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import levy_l
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)

    Calculate the first four moments:

    >>> mean, var, skew, kurt = levy_l.stats(moments='mvsk')

    Display the probability density function (``pdf``):

    >>> # `levy_l` is very heavy-tailed.
    >>> # To show a nice plot, let's cut off the lower 40 percent.
    >>> a, b = levy_l.ppf(0.4), levy_l.ppf(1)
    >>> x = np.linspace(a, b, 100)
    >>> ax.plot(x, levy_l.pdf(x),
    ...        'r-', lw=5, alpha=0.6, label='levy_l pdf')

    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.

    Freeze the distribution and display the frozen ``pdf``:

    >>> rv = levy_l()
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

    Check accuracy of ``cdf`` and ``ppf``:

    >>> vals = levy_l.ppf([0.001, 0.5, 0.999])
    >>> np.allclose([0.001, 0.5, 0.999], levy_l.cdf(vals))
    True

    Generate random numbers:

    >>> r = levy_l.rvs(size=1000)

    And compare the histogram:

    >>> # manual binning to ignore the tail
    >>> bins = np.concatenate(([np.min(r)], np.linspace(a, b, 20)))
    >>> ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.set_xlim([x[0], x[-1]])
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zlevy_l_gen._shape_info  r¦   r8   c                 ó®   — t        |«      }dt        j                  dt        j                  z  |z  «      z  |z  t        j                  dd|z  z  «      z  S rÊ  )rŽ  rP   rÿ   rñ   r·   ©rE   rq   r  s      r6   rr   zlevy_l_gen._pdf  sF   € ä�‹VˆØ”—‘˜œ2Ÿ5™5™ ™Ó$Ñ$ RÑ'¬¯©¨r°1°R±4©yÓ(9Ñ9Ð9r8   c                 óf   — t        |«      }dt        dt        j                  |«      z  «      z  dz
  S r‹  )rŽ  rÀ   rP   rÿ   rÚ  s      r6   ru   zlevy_l_gen._cdf  s,   € Ü�‹VˆØ”9˜Q¤§¡¨£™_Ó-Ñ-°Ñ1Ð1r8   c                 ó`   — t        |«      }dt        dt        j                  |«      z  «      z  S r‹  )rŽ  rÊ   rP   rÿ   rÚ  s      r6   ry   zlevy_l_gen._sf#  s'   € Ü�‹VˆØ”8˜A¤§¡¨£™OÓ,Ñ,Ð,r8   c                 ó4   — t        |dz   dz  «      }d||z  z  S )Nr‰   rU   r<  rÏ   rÐ  s      r6   r~   zlevy_l_gen._ppf'  s#   € Ü˜˜S™ A™Ó&ˆØ�s˜S‘yÑ!Ð!r8   c                 ó*   — dt        |dz  «      dz  z  S )Nr¿  rU   rê   r  s     r6   r�   zlevy_l_gen._isf+  s   € Ø”)˜A˜a™C“. !Ñ#Ñ#Ð#r8   c                 ó~   — t         j                  t         j                  t         j                  t         j                  fS rN   r  rj   s    r6   r   zlevy_l_gen._stats.  rœ  r8   NrÔ  r‡   r8   r6   r×  r×  Í  s9   „ ñFðN "×4Ñ4€Mòò:ò
2ò-ò"ò$ó.r8   r×  Úlevy_lc                   óŒ   ‡ — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zd„ Zd„ Ze ee«      ˆ fd„«       «       Zˆ xZS )Úlogistic_genaã  A logistic (or Sech-squared) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `logistic` is:

    .. math::

        f(x) = \frac{\exp(-x)}
                    {(1+\exp(-x))^2}

    `logistic` is a special case of `genlogistic` with ``c=1``.

    Remark that the survival function (``logistic.sf``) is equal to the
    Fermi-Dirac distribution describing fermionic statistics.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zlogistic_gen._shape_infoM  r¦   r8   c                 ó&   — |j                  |¬«      S r4  )ÚlogisticrÖ   s      r6   rÙ   zlogistic_gen._rvsP  s   € Ø×$Ñ$¨$Ð$Ó/Ð/r8   c                 óJ   — t        j                  | j                  |«      «      S rN   rÝ  r©   s     r6   rr   zlogistic_gen._pdfS  rÞ  r8   c                 óŠ   — t        j                  |«       }|dt        j                  t        j                  |«      «      z  z
  S r>  )rP   rŽ  rw   r¦  r·   )rE   rq   r�  s      r6   rÞ   zlogistic_gen._logpdfW  s2   € Ü�V‰V�A‹YˆJˆØ�2œŸ™¤§¡¨£Ó+Ñ+Ñ+Ð+r8   c                 ó,   — t        j                  |«      S rN   r4  r©   s     r6   ru   zlogistic_gen._cdf[  ó   € Ü�x‰x˜‹{Ðr8   c                 ó,   — t        j                  |«      S rN   ©rw   Ú	log_expitr©   s     r6   rã   zlogistic_gen._logcdf^  s   € Ü�|‰|˜A‹Ðr8   c                 ó,   — t        j                  |«      S rN   r8  r°   s     r6   r~   zlogistic_gen._ppfa  ré  r8   c                 ó.   — t        j                  | «      S rN   r4  r©   s     r6   ry   zlogistic_gen._sfd  s   € Ü�x‰x˜˜‹|Ðr8   c                 ó.   — t        j                  | «      S rN   rë  r©   s     r6   rç   zlogistic_gen._logsfg  s   € Ü�|‰|˜Q˜BÓÐr8   c                 ó.   — t        j                  |«       S rN   r8  r°   s     r6   r�   zlogistic_gen._isfj  s   € Ü—‘˜“ˆ|Ðr8   c                 óR   — dt         j                  t         j                  z  dz  ddfS )Nr   rD  g333333ó?r/  rj   s    r6   r   zlogistic_gen._statsm  s!   € Ø”"—%‘%œŸ™‘+˜c‘/ 1 gÐ-Ð-r8   c                  ó   — yr>  r‡   rj   s    r6   rò   zlogistic_gen._entropyp  s   € àr8   c                 óŠ  •‡‡
‡‡— |j                  dd«      rt        ‰| �  ‰g|¢­i |¤ŽS t        | ‰||«      \  Š}}t	        ‰«      Š| j                  ‰«      \  }}|j                  d|«      |j                  d|«      }}|fˆˆfd„	Š
|fˆˆfd„	Šˆ
ˆfd„}|�+|€)t        j                  ‰
|f«      }	|	j                  d   }|}nT|�+|€)t        j                  ‰|f«      }	|	j                  d   }|}n't        j                  |||f«      }	|	j                  \  }}t        |«      }|	j                  r||fS t        ‰| �  ‰g|¢­i |¤ŽS )	Nr;  Fr.   r/   c                 óp   •— ‰| z
  |z  }t        j                  t        j                  |«      «      ‰dz  z
  S r  )rP   r¥  rw   r5  )r.   r/   r  rF   rb   s      €€r6   Údl_dlocz!logistic_gen.fit.<locals>.dl_dlocˆ  s1   ø€ Ø˜‘˜uÑ$ˆAÜ—6‘6œ"Ÿ(™( 1›+Ó&¨¨1©Ñ,Ð,r8   c                 óv   •— ‰|z
  | z  }t        j                  |t        j                  |dz  «      z  «      ‰z
  S r  )rP   r¥  r/  )r/   r.   r  rF   rb   s      €€r6   Ú	dl_dscalez#logistic_gen.fit.<locals>.dl_dscaleŒ  s5   ø€ Ø˜‘˜uÑ$ˆAÜ—6‘6˜!œBŸG™G A a¡C›L™.Ó)¨AÑ-Ð-r8   c                 ó2   •— | \  }} ‰||«       ‰||«      fS rN   r‡   )Úparamsr.   r/   rõ  r÷  s      €€r6   r^  zlogistic_gen.fit.<locals>.func�  s%   ø€ Ø‰JˆC�Ù˜3 Ó&©	°%¸Ó(=Ð=Ð=r8   r   )r3   rA   rC   rG  r¤  r•  r=   r   rH  rq   rŽ  Úsuccess)rE   rF   rG   r5   rö   r÷   r.   r/   r^  rõ  rõ  r÷  rb   r–  s    `        @@@€r6   rC   zlogistic_gen.fitt  sS  ü€ ð �8‰8�J Ô&Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ä8¸¸tØ9=¸tóEÑˆˆd�Fä�‹Iˆð —^‘^ DÓ)‰
ˆˆUà—X‘X˜e SÓ)¨4¯8©8°G¸UÓ+CˆUˆð  &ö 	-ð "&ö 	.õ	>ð Ð $ ,Ü—-‘- ¨#¨Ó0ˆCØ—%‘%˜‘(ˆCØ‰EØÐ & .Ü—-‘- 	¨E¨8Ó4ˆCØ—E‘E˜!‘HˆEØ‰Cä—-‘-  s¨E lÓ3ˆCØŸ™‰JˆC�ô �E“
ˆØ #§¢��e�ð 	7Ü‘W‘[ Ð5¨Ò5°Ñ5ð	7r8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   rã   r~   ry   rç   r�   r   rò   rK   r   r   rC   rÐ  rÑ  s   @r6   râ  râ  5  se   ø„ ñò.ó0ò'ò,òòòòò òò.òð Ù˜MÓ*ó07ó +ó ô07r8   râ  rå  c                   óN   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Zd„ Zy)Úloggamma_gena½  A log gamma continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `loggamma` is:

    .. math::

        f(x, c) = \frac{\exp(c x - \exp(x))}
                       {\Gamma(c)}

    for all :math:`x, c > 0`. Here, :math:`\Gamma` is the
    gamma function (`scipy.special.gamma`).

    `loggamma` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zloggamma_gen._shape_infoÅ  r  r8   Nc                 ó¦   — t        j                  |j                  |dz   |¬«      «      t        j                  |j                  |¬«      «      |z  z   S )Nr   r  )rP   rð   rØ  r  rë  s       r6   rÙ   zloggamma_gen._rvsÈ  sM   € ô —‘�|×)Ñ)¨!¨a©%°dÐ)Ó;Ó<Ü—&‘&˜×-Ñ-°4Ð-Ó8Ó9¸!Ñ;ñ<ð 	=r8   c                 óŠ   — t        j                  ||z  t        j                  |«      z
  t        j                  |«      z
  «      S rN   ©rP   r·   rw   rÆ  r
  s      r6   rr   zloggamma_gen._pdfÖ  s.   € ä�v‰v�a˜‘cœ"Ÿ&™& ›)‘m¤B§J¡J¨q£MÑ1Ó2Ð2r8   c                 ód   — ||z  t        j                  |«      z
  t        j                  |«      z
  S rN   r   r
  s      r6   rÞ   zloggamma_gen._logpdfÚ  s%   € Ø�‰s”R—V‘V˜A“Y‰¤§¡¨A£Ñ.Ð.r8   c                 ó6   — t        |t        k  ||fd„ d„ ¬«      S )Nc                 ód   — t        j                  || z  t        j                  |dz   «      z
  «      S r^   r   r£  s     r6   rç  z#loggamma_gen._cdf.<locals>.<lambda>í  s$   € ¤r§v¡v¨a°©c´B·J±J¸qÀ¹s³OÑ.CÓ'D€ r8   c                 óT   — t        j                  |t        j                  | «      «      S rN   )rw   r½  rP   r·   r£  s     r6   rç  z#loggamma_gen._cdf.<locals>.<lambda>î  s   € ¬"¯+©+°a¼¿¹À»Ó*C€ r8   rê  ©r   r#   r
  s      r6   ru   zloggamma_gen._cdfÝ  s%   € ô ˜!œh™,¨¨A¨ÙDÙCôEð 	Er8   c                 ód   — t        j                  ||«      }t        |t        k  |||fd„ d„ ¬«      S )Nc                 ód   — t        j                  |«      t        j                  |dz   «      z   |z  S r^   r|  ©rõ  r}   r  s      r6   rç  z#loggamma_gen._ppf.<locals>.<lambda>õ  s$   € ¬2¯6©6°!«9´r·z±zÀ!ÀAÁ#³Ñ+FÈÑ*I€ r8   c                 ó,   — t        j                  | «      S rN   r2  r  s      r6   rç  z#loggamma_gen._ppf.<locals>.<lambda>ö  ó   € ¬R¯V©V°A«Y€ r8   rê  )rw   rÄ  r   r"   ©rE   r}   r  rõ  s       r6   r~   zloggamma_gen._ppfð  s5   € ô �N‰N˜1˜aÓ ˆÜ˜!œe™) a¨¨A YÙIÙ6ô8ð 	8r8   c                 ó6   — t        |t        k  ||fd„ d„ ¬«      S )Nc                 óf   — t        j                  || z  t        j                  |dz   «      z
  «       S r^   )rP   r  rw   rÆ  r£  s     r6   rç  z"loggamma_gen._sf.<locals>.<lambda>û  s'   € ¬¯©°°1±´r·z±zÀ!ÀAÁ#³Ñ1FÓ(GÐ'G€ r8   c                 óT   — t        j                  |t        j                  | «      «      S rN   )rw   rÀ  rP   r·   r£  s     r6   rç  z"loggamma_gen._sf.<locals>.<lambda>ü  s   € ¬"¯,©,°q¼"¿&¹&À»)Ó*D€ r8   rê  r  r
  s      r6   ry   zloggamma_gen._sfø  s#   € ä˜!œh™,¨¨A¨ÙGÙDôFð 	Fr8   c                 ód   — t        j                  ||«      }t        |t        k  |||fd„ d„ ¬«      S )Nc                 óf   — t        j                  | «      t        j                  |dz   «      z   |z  S r^   )rP   r¦  rw   rÆ  r  s      r6   rç  z#loggamma_gen._isf.<locals>.<lambda>  s&   € ¬2¯8©8°Q°B«<¼"¿*¹*ÀQÀqÁS»/Ñ+IÈ1Ñ*L€ r8   c                 ó,   — t        j                  | «      S rN   r2  r  s      r6   rç  z#loggamma_gen._isf.<locals>.<lambda>  r
  r8   rê  )rw   rÈ  r   r"   r  s       r6   r�   zloggamma_gen._isfþ  s5   € ô �O‰O˜A˜qÓ!ˆÜ˜!œe™) a¨¨A YÙLÙ6ô8ð 	8r8   c                 óö   — t        j                  |«      }t        j                  d|«      }t        j                  d|«      t        j                  |d«      z  }t        j                  d|«      ||z  z  }||||fS )Nr   rU   rÊ  r†  )rw   rÐ  Ú	polygammarP   rÌ  )rE   r  rþ   r§  r.  Úexcess_kurtosiss         r6   r   zloggamma_gen._stats  si   € ô �z‰z˜!‹}ˆÜ�l‰l˜1˜aÓ ˆÜ—<‘<  1Ó%¬¯©°°cÓ(:Ñ:ˆÜŸ,™, q¨!Ó,°°C±Ñ8ˆØ�S˜( OÐ3Ð3r8   c                 ó8   — d„ }d„ }t        |dk\  |f||¬«      }|S )Nc                 óh   — t        j                  | «      | t        j                  | «      z  z
  | z   }|S rN   )rw   rÆ  rÐ  )r  ró  s     r6   r¯  z&loggamma_gen._entropy.<locals>.regular  s+   € Ü—
‘
˜1“ ¤B§J¡J¨q£MÑ 1Ñ1°AÑ5ˆAØˆHr8   c                 óš   — dt        j                  | «      z  | dz  dz  z   | dz  dz  z
  | dz  dz  z   }t        j                  «       |z   }|S )Nr$  r<  r…  r´  r  rÁ  éÒ   )rP   rð   r  rò   )r  Útermró  s      r6   rp  z)loggamma_gen._entropy.<locals>.asymptotic  sO   € àœŸ™˜q›	‘> A s¡F¨1¡HÑ,¨q°#©v°b©yÑ8¸1¸c¹6À#¹:ÑEˆDÜ—‘“ $Ñ&ˆAØˆHr8   é-   rŒ  rì  )rE   r  r¯  rp  ró  s        r6   rò   zloggamma_gen._entropy  s)   € ò	ò	ô �q˜B‘w  ¨¸Ô@ˆØˆr8   r  ©rƒ   r„   r…   r†   rk   rÙ   rr   rÞ   ru   r~   ry   r�   r   rò   r‡   r8   r6   rü  rü  ¬  s<   „ ñò0Eó=ò3ò/òEò&8òFò8ò4ór8   rü  Úloggammac                   ór   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Ze ee«      ˆ fd
„«       «       Zˆ xZS )Úloglaplace_genaT  A log-Laplace continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `loglaplace` is:

    .. math::

        f(x, c) = \begin{cases}\frac{c}{2} x^{ c-1}  &\text{for } 0 < x < 1\\
                               \frac{c}{2} x^{-c-1}  &\text{for } x \ge 1
                  \end{cases}

    for :math:`c > 0`.

    `loglaplace` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    Suppose a random variable ``X`` follows the Laplace distribution with
    location ``a`` and scale ``b``.  Then ``Y = exp(X)`` follows the
    log-Laplace distribution with ``c = 1 / b`` and ``scale = exp(a)``.

    References
    ----------
    T.J. Kozubowski and K. Podgorski, "A log-Laplace growth rate model",
    The Mathematical Scientist, vol. 28, pp. 49-60, 2003.

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zloglaplace_gen._shape_infoB  r  r8   c                 óX   — |dz  }t        j                  |dk  || «      }|||dz
  z  z  S r  ©rP   rO  )rE   rq   r  Úcd2s       r6   rr   zloglaplace_gen._pdfE  s7   € ð �‰eˆÜ�H‰H�Q˜‘U˜A ˜rÓ"ˆØ�1�q˜‘s‘8‰|Ðr8   c                 óV   — t        j                  |dk  d||z  z  dd|| z  z  z
  «      S ©Nr   r”   r!  r
  s      r6   ru   zloglaplace_gen._cdfL  s/   € Ü�x‰x˜˜A™˜s 1 a¡4™x¨¨3¨q°A°2©w©;©Ó7Ð7r8   c                 óV   — t        j                  |dk  dd||z  z  z
  d|| z  z  «      S r$  r!  r
  s      r6   ry   zloglaplace_gen._sfO  s/   € Ü�x‰x˜˜A™˜q 3 q¨!¡t¡8™|¨S°°a°R±©[Ó9Ð9r8   c                 ó`   — t        j                  |dk  d|z  d|z  z  dd|z
  z  d|z  z  «      S ©Nr”   r¶   r‰   rU   r<  r!  r  s      r6   r~   zloglaplace_gen._ppfR  s7   € Ü�x‰x˜˜C™ # a¡%¨3¨q©5Ñ!1°A°s¸1±u±IÀÀaÁÑ3HÓIÐIr8   c                 ó`   — t        j                  |dkD  dd|z
  z  d|z  z  d|z  d|z  z  «      S r'  r!  r  s      r6   r�   zloglaplace_gen._isfU  s6   € Ü�x‰x˜˜C™ # s¨Q¡w¡-°3°q±5Ñ!9¸A¸a¹CÀ4ÈÁ6¹?ÓKÐKr8   c                 óÊ   — t        j                  d¬«      5  |dz  |dz  }}t        j                  ||k  |||z
  z  t         j                  «      cd d d «       S # 1 sw Y   y xY w)Nr9  r:  rU   )rP   r<  rO  ri   )rE   rb   r  rß  Ún2s        r6   r  zloglaplace_gen._munpX  sT   € Ü�[‰[ Ô)ñ 	=Ø˜‘T˜1˜a™4�ˆBÜ—8‘8˜B ™G R¨2°©7¡^´R·V±VÓ<÷	=÷ 	=ò 	=ús   —8AÁA"c                 ó8   — t        j                  d|z  «      dz   S r\  r2  r~  s     r6   rò   zloglaplace_gen._entropy]  s   € Ü�v‰v�c˜!‘e‹}˜sÑ"Ð"r8   c                 óÎ  •— t        | |||«      \  }}}}|€t        t        | «      | �  |g|¢­i |¤ŽS t	        j
                  ||k  «      rt        d|t        j                  ¬«      ‚|dk7  r||z
  }t        j                  t	        j                  |«      |�t	        j                  |«      nd |�d|z  nd d¬«      \  }}|}	|€t	        j                  |«      n|}
|€d|z  n|}||	|
fS )NÚ
loglaplacerŸ  r   r   r;   )rö   r÷   r1   )rG  rA   rB   rC   rP   r£  rK  ri   r’  rð   r·   )rE   rF   rG   r5   rI  rö   r÷   r‹   rŒ   r.   r/   r  r–  s               €r6   rC   zloglaplace_gen.fit`  sú   ø€ ô "=¸TÀ4Ø=AÀ4ó"IÑˆˆb�$˜ð ˆ<Üœ˜d› TÑ.¨tÐC°dÒC¸dÑCÐCô �6‰6�$˜$‘,ÔÜ˜|°4¼r¿v¹vÔFÐFð �1Š9Ø˜$‘;ˆDô �{‰{œ2Ÿ6™6 $›<Ø28Ð2D¤§¡ v¤È$Ø*,¨. ! B¢$¸dØ"'ð ó )‰ˆˆ1ð ˆØ#˜^”—‘�q”	°ˆØ�ZˆA�ŠE RˆØ�#�uˆ}Ðr8   )rƒ   r„   r…   r†   rk   rr   ru   ry   r~   r�   r  rò   rK   r   r   rC   rÐ  rÑ  s   @r6   r  r  !  sU   ø„ ñò@Eòò8ò:òJòLò=ò
#ð Ù˜MÓ*óó +ó ôr8   r  r-  c                 óH   — t        | dk7  | |fd„ t        j                   «      S )Nr   c                 óÆ   — t        j                  | «      dz   d|dz  z  z  t        j                  || z  t        j                  dt         j                  z  «      z  «      z
  S r  )rP   rð   rÿ   rñ   ©rq   r  s     r6   rç  z!_lognorm_logpdf.<locals>.<lambda>†  sM   € ¤R§V¡V¨A£Y°¡\ M°Q¸¸A¹±XÑ$>Ü&(§f¡f¨Q°©U´R·W±W¸QÄÇÁ¹YÓ5GÑ-GÓ&Hñ%I€ r8   r  r0  s     r6   Ú_lognorm_logpdfr1  „  s*   € Ü�a˜1‘f˜q !˜fñJä—v‘v�góð r8   c                   ó¨   ‡ — e Zd ZdZej
                  Zd„ Zdd„Zd„ Z	d„ Z
d„ Zd„ Zd„ Zd	„ Zd
„ Zd„ Zd„ Zd„ Ze eed¬«      ˆ fd„«       «       Zˆ xZS )Úlognorm_gena±  A lognormal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `lognorm` is:

    .. math::

        f(x, s) = \frac{1}{s x \sqrt{2\pi}}
                  \exp\left(-\frac{\log^2(x)}{2s^2}\right)

    for :math:`x > 0`, :math:`s > 0`.

    `lognorm` takes ``s`` as a shape parameter for :math:`s`.

    %(after_notes)s

    Suppose a normally distributed random variable ``X`` has  mean ``mu`` and
    standard deviation ``sigma``. Then ``Y = exp(X)`` is lognormally
    distributed with ``s = sigma`` and ``scale = exp(mu)``.

    %(example)s

    The logarithm of a log-normally distributed random variable is
    normally distributed:

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy import stats
    >>> fig, ax = plt.subplots(1, 1)
    >>> mu, sigma = 2, 0.5
    >>> X = stats.norm(loc=mu, scale=sigma)
    >>> Y = stats.lognorm(s=sigma, scale=np.exp(mu))
    >>> x = np.linspace(*X.interval(0.999))
    >>> y = Y.rvs(size=10000)
    >>> ax.plot(x, X.pdf(x), label='X (pdf)')
    >>> ax.hist(np.log(y), density=True, bins=x, label='log(Y) (histogram)')
    >>> ax.legend()
    >>> plt.show()

    c                 ó@   — t        dddt        j                  fd«      gS )Nr  Fr   r
  rh   rj   s    r6   rk   zlognorm_gen._shape_info¹  r  r8   c                 óP   — t        j                  ||j                  |«      z  «      S rN   ©rP   r·   rÕ   )rE   r  r×   rØ   s       r6   rÙ   zlognorm_gen._rvs¼  s!   € Ü�v‰v�a˜,×6Ñ6°tÓ<Ñ<Ó=Ð=r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  ©rE   rq   r  s      r6   rr   zlognorm_gen._pdf¿  r½  r8   c                 ó   — t        ||«      S rN   ©r1  r8  s      r6   rÞ   zlognorm_gen._logpdfÃ  s   € Ü˜q !Ó$Ð$r8   c                 óD   — t        t        j                  |«      |z  «      S rN   ©rÀ   rP   rð   r8  s      r6   ru   zlognorm_gen._cdfÆ  s   € ÜœŸ™ › Q™Ó'Ð'r8   c                 óD   — t        t        j                  |«      |z  «      S rN   r”  r8  s      r6   rã   zlognorm_gen._logcdfÉ  s   € ÜœBŸF™F 1›I¨™MÓ*Ð*r8   c                 óD   — t        j                  |t        |«      z  «      S rN   ©rP   r·   rÇ   ©rE   r}   r  s      r6   r~   zlognorm_gen._ppfÌ  ó   € Ü�v‰v�aœ) A›,Ñ&Ó'Ð'r8   c                 óD   — t        t        j                  |«      |z  «      S rN   ©rÊ   rP   rð   r8  s      r6   ry   zlognorm_gen._sfÏ  s   € ÜœŸ™˜q›	 A™Ó&Ð&r8   c                 óD   — t        t        j                  |«      |z  «      S rN   )rÍ   rP   rð   r8  s      r6   rç   zlognorm_gen._logsfÒ  s   € Üœ2Ÿ6™6 !›9 q™=Ó)Ð)r8   c                 óD   — t        j                  |t        |«      z  «      S rN   ©rP   r·   rÐ   r@  s      r6   r�   zlognorm_gen._isfÕ  rA  r8   c                 óä   — t        j                  ||z  «      }t        j                  |«      }||dz
  z  }t        j                  |dz
  «      d|z   z  }t        j                  g d¢|«      }||||fS ©Nr   rU   )r   rU   r†  r   rÃ  )rP   r·   rÿ   Úpolyval)rE   r  rô  rC  rD  rE  rF  s          r6   r   zlognorm_gen._statsØ  sf   € Ü�F‰F�1�Q‘3‹KˆÜ�W‰W�Q‹ZˆØ��1‘‰gˆÜ�W‰W�Q�q‘S‹\˜1˜Q™3ÑˆÜ�Z‰ZÒ*¨AÓ.ˆØ�3˜˜BˆÐr8   c                 óŒ   — ddt        j                  dt         j                  z  «      z   dt        j                  |«      z  z   z  S ©Nr”   r   rU   rï   )rE   r  s     r6   rò   zlognorm_gen._entropyà  s3   € Ø�aœ"Ÿ&™& ¤2§5¡5¡›/Ñ)¨A´·±°q³	©MÑ9Ñ:Ð:r8   aF          When `method='MLE'` and
        the location parameter is fixed by using the `floc` argument,
        this function uses explicit formulas for the maximum likelihood
        estimation of the log-normal shape and scale parameters, so the
        `optimizer`, `loc` and `scale` keyword arguments are ignored.
        If the location is free, a likelihood maximum is found by
        setting its partial derivative wrt to location to 0, and
        solving by substituting the analytical expressions of shape
        and scale (or provided parameters).
        See, e.g., equation 3.1 in
        A. Clifford Cohen & Betty Jones Whitten (1980)
        Estimation in the Three-Parameter Lognormal Distribution,
        Journal of the American Statistical Association, 75:370, 399-404
        https://doi.org/10.2307/2287466
        

ró   c                 óÞ  •‡ ‡‡‡‡— |j                  dd«      rt        ‰‰ �  ‰g|¢­i |¤ŽS t        ‰ ‰||«      }|\  ŠŠ}Št	        j
                  ‰«      }ˆˆˆfd„Šˆˆfd„}ˆˆˆ fd„}|�€&t	        j                  |«      }	||	z
  }
 ||
«      } ||
«      }d|	z  }|dk\  r||z
  }
 ||
«      }|dz  }|dk\  rŒt	        j                  |
«      rt	        j                  |«      st        ‰‰ �  ‰g|¢­i |¤ŽS t	        j                  t	        j                  |
t        j                   «      |
dz
  «      } ||«      }d|
|z
  z  }t	        j                  |«      r¨t	        j                  |«      r“t	        j                  |«      t	        j                  |«      k(  rh|
|z
  } ||«      }|dz  }t	        j                  |«      rAt	        j                  |«      r,t	        j                  |«      t	        j                  |«      k(  rŒht	        j                  |«      rt	        j                  |«      st        ‰‰ �  ‰g|¢­i |¤ŽS t        |||
f¬	«      }|j                  st        ‰‰ �  ‰g|¢­i |¤ŽS  ||j                  «      }||kD  r|j                  n||	z
  }n#||k\  rt        d
dt        j                  ¬«      ‚|} ‰|«      \  }}‰ j!                  |«      r|dkD  st        ‰‰ �  ‰g|¢­i |¤ŽS |||fS )Nr;  Fc                 ó  •— ‰�‰€t        j                  ‰| z
  «      }‰xs# t        j                  j                  «       «      }‰xsA t        j                  t        j                  t        j                  |«      z
  dz  «      «      }||fS r  )rP   rð   r·   rþ   rÿ   )r.   Úlndatar/   r·  rF   r÷   Úfshapes       €€€r6   Úget_shape_scalez(lognorm_gen.fit.<locals>.get_shape_scaleü  sq   ø€ ð ˆ~  ÜŸ™  s¡
Ó+�ØÒ3œbŸf™f V§[¡[£]Ó3ˆEØÒKœbŸg™g¤b§g¡g¨v¼¿¹¸u»Ñ/EÈÑ.IÓ&JÓKˆEØ˜%�<Ðr8   c                 ó’   •—  ‰| «      \  }}‰| z
  }t        j                  dt        j                  ||z  «      |dz  z  z   |z  «      S r1  ©rP   r¥  rð   )r.   r·  r/   ÚshiftedrF   rP  s       €€r6   ÚdL_dLocz lognorm_gen.fit.<locals>.dL_dLoc  sI   ø€ á*¨3Ó/‰LˆE�5Ø˜S‘jˆGÜ—6‘6˜1œrŸv™v g¨e¡mÓ4°U¸A±XÑ=Ñ=¸wÑFÓGÐGr8   c                 óF   •—  ‰| «      \  }}‰j                  || |f‰«       S rN   )Únnlf)r.   r·  r/   rF   rP  rE   s      €€€r6   Úllzlognorm_gen.fit.<locals>.ll  s,   ø€ á*¨3Ó/‰LˆE�5Ø—I‘I˜u c¨5Ð1°4Ó8Ð8Ð8r8   rU   g�íµ ÷Æ°¾r   r!  Úlognormrˆ   rŸ  r   )r3   rA   rC   rG  rP   r‡  Úspacingrü   rØ  Ú	nextafterri   rQ   r+   Ú	convergedrH  rK  rc   )rE   rF   rG   r5   Ú
parametersrö   rˆ  rT  rW  rY  rS   ÚdL_dLoc_rbrackÚ	ll_rbrackr%  rR   ÚdL_dLoc_lbrackrõ  Úll_rootr.   r·  r/   r÷   rO  rP  r–  s   ``                   @@@€r6   rC   zlognorm_gen.fitã  s·  ý€ ð$ �8‰8�J Ô&Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ä0°°t¸TÀ4ÓHˆ
Ø%/Ñ"ˆˆf�d˜FÜ—6‘6˜$“<ˆö	 õ	Hö	9ð
 ‰<ô —j‘j Ó*ˆGØ Ñ'ˆFñ % V›_ˆNÙ˜6›
ˆIØ˜‘KˆEØ  EÒ)Ø! EÑ)�Ù!(¨£�Ø˜‘
�ð ! EÓ)ô
 —;‘;˜vÔ&¬b¯k©k¸.Ô.Iô ‘w‘{ 4Ð7¨$Ò7°$Ñ7Ð7ô
 —Z‘Z¤§¡¨V´b·f±f°WÓ =¸vÀa¹xÓHˆFÙ$ V›_ˆNØ˜ &™Ñ)ˆEÜ—;‘;˜vÔ&¬2¯;©;°~Ô+FÜ—w‘w˜~Ó.´"·'±'¸.Ó2IÒIØ %™�Ù!(¨£�Ø˜‘
�ô	 —;‘;˜vÔ&¬2¯;©;°~Ô+FÜ—w‘w˜~Ó.´"·'±'¸.Ó2IÓIô —;‘;˜vÔ&¬b¯k©k¸.Ô.IÜ‘w‘{ 4Ð7¨$Ò7°$Ñ7Ð7ô ˜g°¸Ð/?Ô@ˆCØ—=’=Ü‘w‘{ 4Ð7¨$Ò7°$Ñ7Ð7ñ
 ˜Ÿ™“lˆGØ%¨	Ò1�#—(’(°xÀÑ7G‰Cð �xÒÜ" 9°B¼b¿f¹fÔEÐEØˆCá& sÓ+‰ˆˆuØ—‘˜uÔ%¨%°!ª)Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3Ø�c˜5Ð Ð r8   r  )rƒ   r„   r…   r†   r   r  r  rk   rÙ   rr   rÞ   ru   rã   r~   ry   rç   r�   r   rò   rK   r	   rC   rÐ  rÑ  s   @r6   r3  r3  ‹  s}   ø„ ñ*ðV "×4Ñ4€MòEó>ò*ò%ò(ò+ò(ò'ò*ò(òò;ð Ù˜}ð 5ô ó Z!ó!ó ô"Z!r8   r3  rX  c                   óf   — e Zd ZdZej
                  Zd„ Zdd„Zd„ Z	d„ Z
d„ Zd„ Zd	„ Zd
„ Zd„ Zd„ Zy)Ú
gibrat_gena[  A Gibrat continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `gibrat` is:

    .. math::

        f(x) = \frac{1}{x \sqrt{2\pi}} \exp(-\frac{1}{2} (\log(x))^2)

    for :math:`x >= 0`.

    `gibrat` is a special case of `lognorm` with ``s=1``.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zgibrat_gen._shape_infol  r¦   r8   Nc                 óJ   — t        j                  |j                  |«      «      S rN   r6  rÖ   s      r6   rÙ   zgibrat_gen._rvso  s   € Ü�v‰v�l×2Ñ2°4Ó8Ó9Ð9r8   c                 óJ   — t        j                  | j                  |«      «      S rN   rÝ  r©   s     r6   rr   zgibrat_gen._pdfr  rÞ  r8   c                 ó   — t        |d«      S r  r:  r©   s     r6   rÞ   zgibrat_gen._logpdfv  s   € Ü˜q #Ó&Ð&r8   c                 ó>   — t        t        j                  |«      «      S rN   r<  r©   s     r6   ru   zgibrat_gen._cdfy  s   € ÜœŸ™ ›Ó#Ð#r8   c                 ó>   — t        j                  t        |«      «      S rN   r?  r°   s     r6   r~   zgibrat_gen._ppf|  ó   € Ü�v‰v”i “lÓ#Ð#r8   c                 ó>   — t        t        j                  |«      «      S rN   rC  r©   s     r6   ry   zgibrat_gen._sf  s   € ÜœŸ™˜q›	Ó"Ð"r8   c                 ó>   — t        j                  t        |«      «      S rN   rF  r  s     r6   r�   zgibrat_gen._isf‚  ri  r8   c                 óÔ   — t         j                  }t        j                  |«      }||dz
  z  }t        j                  |dz
  «      d|z   z  }t        j                  g d¢|«      }||||fS rH  )rP   Úerÿ   rI  )rE   rô  rC  rD  rE  rF  s         r6   r   zgibrat_gen._stats…  s^   € Ü�D‰DˆÜ�W‰W�Q‹ZˆØ�1�q‘5‰kˆÜ�W‰W�Q˜‘U‹^˜q 1™uÑ%ˆÜ�Z‰ZÒ*¨AÓ.ˆØ�3˜˜BˆÐr8   c                 óZ   — dt        j                  dt         j                  z  «      z  dz   S r»  rï   rj   s    r6   rò   zgibrat_gen._entropy�  s#   € Ø”R—V‘V˜A¤§¡™IÓ&Ñ&¨Ñ,Ð,r8   r  )rƒ   r„   r…   r†   r   r  r  rk   rÙ   rr   rÞ   ru   r~   ry   r�   r   rò   r‡   r8   r6   rb  rb  T  sF   „ ñð* "×4Ñ4€Mòó:ò'ò'ò$ò$ò#ò$òó-r8   rb  Úgibratc                   óN   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Zd„ Zy)Úmaxwell_gena  A Maxwell continuous random variable.

    %(before_notes)s

    Notes
    -----
    A special case of a `chi` distribution,  with ``df=3``, ``loc=0.0``,
    and given ``scale = a``, where ``a`` is the parameter used in the
    Mathworld description [1]_.

    The probability density function for `maxwell` is:

    .. math::

        f(x) = \sqrt{2/\pi}x^2 \exp(-x^2/2)

    for :math:`x >= 0`.

    %(after_notes)s

    References
    ----------
    .. [1] http://mathworld.wolfram.com/MaxwellDistribution.html

    %(example)s
    c                 ó   — g S rN   r‡   rj   s    r6   rk   zmaxwell_gen._shape_info¯  r¦   r8   Nc                 ó2   — t         j                  d||¬«      S )NrD  r×  ©rÙ  rÙ  rÖ   s      r6   rÙ   zmaxwell_gen._rvs²  s   € Ü�w‰w�s °LˆwÓAÐAr8   c                 óT   — t         |z  |z  t        j                  | |z  dz  «      z  S r>  )r'   rP   r·   r©   s     r6   rr   zmaxwell_gen._pdfµ  s*   € ä˜qÑ  Ñ"¤2§6¡6¨1¨"¨Q©$¨s©(Ó#3Ñ3Ð3r8   c                 óª   — t        j                  d¬«      5  t        dt        j                  |«      z  z   d|z  |z  z
  cd d d «       S # 1 sw Y   y xY w)Nr9  r:  rU   r”   )rP   r<  r)   rð   r©   s     r6   rÞ   zmaxwell_gen._logpdf¹  sD   € ä�[‰[ Ô)ñ 	?Ü&¨¬2¯6©6°!«9©Ñ4°s¸1±u¸Q±wÑ>÷	?÷ 	?ò 	?ús   —(A	Á	Ac                 ó:   — t        j                  d||z  dz  «      S ©NrÊ  r¶   r¼  r©   s     r6   ru   zmaxwell_gen._cdf¾  s   € Ü�{‰{˜3  !¡ C¡Ó(Ð(r8   c                 óZ   — t        j                  dt        j                  d|«      z  «      S rÿ  rÃ  r°   s     r6   r~   zmaxwell_gen._ppfÁ  s!   € Ü�w‰w�qœŸ™¨¨QÓ/Ñ/Ó0Ð0r8   c                 ó:   — t        j                  d||z  dz  «      S rx  r¿  r©   s     r6   ry   zmaxwell_gen._sfÄ  s   € Ü�|‰|˜C  1¡ S¡Ó)Ð)r8   c                 óZ   — t        j                  dt        j                  d|«      z  «      S rÿ  rÇ  r°   s     r6   r�   zmaxwell_gen._isfÇ  s!   € Ü�w‰w�qœŸ™¨¨aÓ0Ñ0Ó1Ð1r8   c                 óŽ  — dt         j                  z  dz
  }dt        j                  dt         j                  z  «      z  ddt         j                  z  z
  t        j                  d«      ddt         j                  z  z
  z  |dz  z  dt         j                  z  t         j                  z  d	t         j                  z  z   d
z
  |dz  z  fS )Nr†  r.  rU   r¶   é    r¶  rÊ  r  é    i€  ©rP   rñ   rÿ   ©rE   rx  s     r6   r   zmaxwell_gen._statsÊ  sš   € Ø”—‘‰g�a‰iˆØ”"—'‘'˜#œbŸe™e™)Ó$Ñ$Ø�!”B—E‘E‘'‘	Ü—‘˜“
˜B˜r¤"§%¡%™x™KÑ(¨¨c©Ñ1Ø”R—U‘U‘œ2Ÿ5™5‘ 3¤r§u¡u¡9Ñ,¨sÑ2°c¸3±hÑ>ð@ð 	@r8   c                 óh   — t         dt        j                  dt        j                  z  «      z  z   dz
  S r»  )r$   rP   rð   rñ   rj   s    r6   rò   zmaxwell_gen._entropyÑ  s'   € Ü˜œBŸF™F 1¤R§U¡U¡7›OÑ+Ñ+¨CÑ/Ð/r8   r  r  r‡   r8   r6   rq  rq  ”  s;   „ ñò4óBò4ò?ò
)ò1ò*ò2ò@ó0r8   rq  Úmaxwellc                   ó4   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	y)	Ú
mielke_genaâ  A Mielke Beta-Kappa / Dagum continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `mielke` is:

    .. math::

        f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}}

    for :math:`x > 0` and :math:`k, s > 0`. The distribution is sometimes
    called Dagum distribution ([2]_). It was already defined in [3]_, called
    a Burr Type III distribution (`burr` with parameters ``c=s`` and
    ``d=k/s``).

    `mielke` takes ``k`` and ``s`` as shape parameters.

    %(after_notes)s

    References
    ----------
    .. [1] Mielke, P.W., 1973 "Another Family of Distributions for Describing
           and Analyzing Precipitation Data." J. Appl. Meteor., 12, 275-280
    .. [2] Dagum, C., 1977 "A new model for personal income distribution."
           Economie Appliquee, 33, 327-367.
    .. [3] Burr, I. W. "Cumulative frequency functions", Annals of
           Mathematical Statistics, 13(2), pp 215-232 (1942).

    %(example)s

    c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS )Nr  Fr   r
  r  rh   )rE   ÚikÚi_ss      r6   rk   zmielke_gen._shape_infoú  ó<   € Ü˜˜U Q¬¯© K°Ó@ˆÜ˜˜e a¬¯© [°.ÓAˆØ�CˆyÐr8   c                 óB   — |||dz
  z  z  d||z  z   d|dz  |z  z   z  z  S r  r‡   ©rE   rq   r  r  s       r6   rr   zmielke_gen._pdfÿ  s2   € Ø��Q�s‘U‘‰|˜s 1 a¡4™x¨3¨q°©u°Q©w©;Ñ7Ñ7Ð7r8   c                 ó   — t        j                  d¬«      5  t        j                  |«      t        j                  |«      |dz
  z  z   t        j                  ||z  «      d||z  z   z  z
  cd d d «       S # 1 sw Y   y xY w)Nr9  r:  r   )rP   r<  rð   r¦  rŠ  s       r6   rÞ   zmielke_gen._logpdf  sf   € ä�[‰[ Ô)ñ 	LÜ—6‘6˜!“9œrŸv™v a›y¨!¨a©%Ñ0Ñ0´2·8±8¸A¸q¹D³>À1ÀqÈÁsÁ7Ñ3KÑK÷	L÷ 	Lò 	Lús   —AA4Á4A=c                 ó0   — ||z  d||z  z   |dz  |z  z  z  S r  r‡   rŠ  s       r6   ru   zmielke_gen._cdf  s&   € Ø�!‰t�s˜1˜a™4‘x 1 S¡5¨¡7Ñ+Ñ+Ð+r8   c                 óP   — t        ||dz  |z  «      }t        |d|z
  z  d|z  «      S r  r4  )rE   r}   r  r  Úqsks        r6   r~   zmielke_gen._ppf
  s.   € Ü�!�Q�s‘U˜1‘W‹oˆÜ�3˜˜C™‘= # a¡%Ó(Ð(r8   c                 óL   — d„ }t        ||k  |||f|t        j                  «      S )Nc                 ó¢   — t        j                  || z   |z  «      t        j                  d| |z  z
  «      z  t        j                  ||z  «      z  S r^   rj  )rb   r  r  s      r6   rT  z$mielke_gen._munp.<locals>.nth_moment  s?   € ä—8‘8˜Q˜q™S !™GÓ$¤R§X¡X¨a°°!±©e£_Ñ4´R·X±X¸aÀ¹c³]ÑBÐBr8   r  )rE   rb   r  r  rT  s        r6   r  zmielke_gen._munp  s)   € ò	Cô ˜!˜a™% ! Q¨ ¨J¼¿¹Ó?Ð?r8   N)
rƒ   r„   r…   r†   rk   rr   rÞ   ru   r~   r  r‡   r8   r6   r„  r„  Ø  s(   „ ñ òBò
8òLò
,ò)ó@r8   r„  Úmielkec                   óR   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zd„ Zy)Ú
kappa4_genap  Kappa 4 parameter distribution.

    %(before_notes)s

    Notes
    -----
    The probability density function for kappa4 is:

    .. math::

        f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}

    if :math:`h` and :math:`k` are not equal to 0.

    If :math:`h` or :math:`k` are zero then the pdf can be simplified:

    h = 0 and k != 0::

        kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
                              exp(-(1.0 - k*x)**(1.0/k))

    h != 0 and k = 0::

        kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0)

    h = 0 and k = 0::

        kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x))

    kappa4 takes :math:`h` and :math:`k` as shape parameters.

    The kappa4 distribution returns other distributions when certain
    :math:`h` and :math:`k` values are used.

    +------+-------------+----------------+------------------+
    | h    | k=0.0       | k=1.0          | -inf<=k<=inf     |
    +======+=============+================+==================+
    | -1.0 | Logistic    |                | Generalized      |
    |      |             |                | Logistic(1)      |
    |      |             |                |                  |
    |      | logistic(x) |                |                  |
    +------+-------------+----------------+------------------+
    |  0.0 | Gumbel      | Reverse        | Generalized      |
    |      |             | Exponential(2) | Extreme Value    |
    |      |             |                |                  |
    |      | gumbel_r(x) |                | genextreme(x, k) |
    +------+-------------+----------------+------------------+
    |  1.0 | Exponential | Uniform        | Generalized      |
    |      |             |                | Pareto           |
    |      |             |                |                  |
    |      | expon(x)    | uniform(x)     | genpareto(x, -k) |
    +------+-------------+----------------+------------------+

    (1) There are at least five generalized logistic distributions.
        Four are described here:
        https://en.wikipedia.org/wiki/Generalized_logistic_distribution
        The "fifth" one is the one kappa4 should match which currently
        isn't implemented in scipy:
        https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution
        https://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html
    (2) This distribution is currently not in scipy.

    References
    ----------
    J.C. Finney, "Optimization of a Skewed Logistic Distribution With Respect
    to the Kolmogorov-Smirnov Test", A Dissertation Submitted to the Graduate
    Faculty of the Louisiana State University and Agricultural and Mechanical
    College, (August, 2004),
    https://digitalcommons.lsu.edu/gradschool_dissertations/3672

    J.R.M. Hosking, "The four-parameter kappa distribution". IBM J. Res.
    Develop. 38 (3), 25 1-258 (1994).

    B. Kumphon, A. Kaew-Man, P. Seenoi, "A Rainfall Distribution for the Lampao
    Site in the Chi River Basin, Thailand", Journal of Water Resource and
    Protection, vol. 4, 866-869, (2012).
    :doi:`10.4236/jwarp.2012.410101`

    C. Winchester, "On Estimation of the Four-Parameter Kappa Distribution", A
    Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March
    2000).
    http://www.nlc-bnc.ca/obj/s4/f2/dsk2/ftp01/MQ57336.pdf

    %(after_notes)s

    %(example)s

    c                 óv   — t        j                  ||«      d   j                  }t        j                  |d¬«      S )Nr   T©Ú
fill_value)rP   rñ  r·  Úfull)rE   ró  r  r·  s       r6   rc   zkappa4_gen._argcheckr  s0   € Ü×#Ñ# A qÓ)¨!Ñ,×2Ñ2ˆÜ�w‰w�u¨Ô.Ð.r8   c                 ó¾   — t        ddt        j                   t        j                  fd«      }t        ddt        j                   t        j                  fd«      }||gS )Nró  Fr
  r  rh   )rE   Úihr†  s      r6   rk   zkappa4_gen._shape_infov  sI   € Ü˜˜U¤b§f¡f W¬b¯f©fÐ$5°~ÓFˆÜ˜˜U¤b§f¡f W¬b¯f©fÐ$5°~ÓFˆØ�Bˆxˆr8   c           
      ó
  — t        j                  |dkD  |dkD  «      t        j                  |dkD  |dk(  «      t        j                  |dkD  |dk  «      t        j                  |dk  |dkD  «      t        j                  |dk  |dk(  «      t        j                  |dk  |dk  «      g}d„ }d„ }d„ }d„ }t        |||||||g||gt         j                  ¬«      }d„ }d„ }t        |||||||g||gt         j                  ¬«      }	||	fS )	Nr   c                 ó<   — dt        j                  | | «      z
  |z  S r  )rP   r­  ©ró  r  s     r6   r™  z#kappa4_gen._get_support.<locals>.f0ƒ  s   € Øœ"Ÿ.™.¨¨Q¨BÓ/Ñ/°Ñ2Ð2r8   c                 ó,   — t        j                  | «      S rN   r2  rœ  s     r6   rœ  z#kappa4_gen._get_support.<locals>.f1†  s   € Ü—6‘6˜!“9Ðr8   c                 ó~   — t        j                  t        j                  | «      «      }t         j                   |d d  |S rN   ©rP   rÓ  r·  ri   ©ró  r  r‹   s      r6   Úf3z#kappa4_gen._get_support.<locals>.f3‰  s,   € Ü—‘œŸ™ !›Ó%ˆAÜ—F‘F�7ˆA‰aˆDØˆHr8   c                 ó   — d|z  S r  r‡   rœ  s     r6   Úf5z#kappa4_gen._get_support.<locals>.f5Ž  ó   € Ø�q‘5ˆLr8   ©Údefaultc                 ó   — d|z  S r  r‡   rœ  s     r6   r™  z#kappa4_gen._get_support.<locals>.f0–  r¤  r8   c                 ó|   — t        j                  t        j                  | «      «      }t         j                  |d d  |S rN   rŸ  r   s      r6   rœ  z#kappa4_gen._get_support.<locals>.f1™  s*   € Ü—‘œŸ™ !›Ó%ˆAÜ—6‘6ˆA‰aˆDØˆHr8   ©rP   rƒ  r   r  )
rE   ró  r  Úcondlistr™  rœ  r¡  r£  rÚ  rÙ  s
             r6   r–   zkappa4_gen._get_support{  s  € Ü—N‘N 1 q¡5¨!¨a©%Ó0Ü—N‘N 1 q¡5¨!¨q©&Ó1Ü—N‘N 1 q¡5¨!¨a©%Ó0Ü—N‘N 1¨¡6¨1¨q©5Ó1Ü—N‘N 1¨¡6¨1°©6Ó2Ü—N‘N 1¨¡6¨1¨q©5Ó1ð3ˆò	3ò	ò	ò
	ô ˜Ø˜b " b¨"¨bÐ1Ø˜Q˜Ü!#§¡ô)ˆò
	ò	ô
 ˜Ø˜b " b¨"¨bÐ1Ø˜Q˜Ü!#§¡ô)ˆð �2ˆvˆr8   c                 óN   — t        j                  | j                  |||«      «      S rN   rÝ  ©rE   rq   ró  r  s       r6   rr   zkappa4_gen._pdf¤  r¬  r8   c                 ó>  — t        j                  |dk7  |dk7  «      t        j                  |dk(  |dk7  «      t        j                  |dk7  |dk(  «      t        j                  |dk(  |dk(  «      g}d„ }d„ }d„ }d„ }t        |||||g|||gt         j                  ¬«      S )Nr   c                 óœ   — t        j                  d|z  dz
  | | z  «      t        j                  d|z  dz
  | d|| z  z
  d|z  z  z  «      z   S )zŒpdf = (1.0 - k*x)**(1.0/k - 1.0)*(
                      1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h-1.0)
               logpdf = ...
            r‰   rf  ©rq   ró  r  s      r6   r™  zkappa4_gen._logpdf.<locals>.f0¯  sZ   € ô
 —J‘J˜s 1™u s™{¨Q¨B¨q©DÓ1Ü—J‘J˜s 1™u s™{¨Q¨B°°a¸±c±	¸SÀ¹UÑ/CÑ,CÓDñEð Fr8   c                 ó`   — t        j                  d|z  dz
  | | z  «      d|| z  z
  d|z  z  z
  S )z~pdf = (1.0 - k*x)**(1.0/k - 1.0)*np.exp(-(
                      1.0 - k*x)**(1.0/k))
               logpdf = ...
            r‰   rf  r¯  s      r6   rœ  zkappa4_gen._logpdf.<locals>.f1·  s9   € ô
 —:‘:˜c !™e c™k¨A¨2¨a©4Ó0°C¸!¸A¹#±IÀÀQÁÑ3GÑGÐGr8   c                 ór   — |  t        j                  d|z  dz
  | t        j                  |  «      z  «      z   S )z]pdf = np.exp(-x)*(1.0 - h*np.exp(-x))**(1.0/h - 1.0)
               logpdf = ...
            r‰   )rw   rw  rP   r·   r¯  s      r6   rë  zkappa4_gen._logpdf.<locals>.f2¾  s4   € ð �2œŸ
™
 3 q¡5¨3¡;°°´2·6±6¸1¸"³:±Ó>Ñ>Ð>r8   c                 ó6   — |  t        j                  |  «      z
  S )zDpdf = np.exp(-x-np.exp(-x))
               logpdf = ...
            ru  r¯  s      r6   r¡  zkappa4_gen._logpdf.<locals>.f3Ä  s   € ð �2œŸ™ ˜r›
‘?Ð"r8   r¥  r©  ©	rE   rq   ró  r  rª  r™  rœ  rë  r¡  s	            r6   rÞ   zkappa4_gen._logpdf©  s¤   € Ü—N‘N 1¨¡6¨1°©6Ó2Ü—N‘N 1¨¡6¨1°©6Ó2Ü—N‘N 1¨¡6¨1°©6Ó2Ü—N‘N 1¨¡6¨1°©6Ó2ð4ˆò
	Fò	Hò	?ò	#ô ˜8Ø  B¨Ð+Ø˜q !˜9Ü#%§6¡6ô+ð 	+r8   c                 óN   — t        j                  | j                  |||«      «      S rN   rO  r¬  s       r6   ru   zkappa4_gen._cdfÏ  rY  r8   c                 ó>  — t        j                  |dk7  |dk7  «      t        j                  |dk(  |dk7  «      t        j                  |dk7  |dk(  «      t        j                  |dk(  |dk(  «      g}d„ }d„ }d„ }d„ }t        |||||g|||gt         j                  ¬«      S )Nr   c                 óX   — d|z  t        j                  | d|| z  z
  d|z  z  z  «      z  S )zVcdf = (1.0 - h*(1.0 - k*x)**(1.0/k))**(1.0/h)
               logcdf = ...
            r‰   r	  r¯  s      r6   r™  zkappa4_gen._logcdf.<locals>.f0Ø  s4   € ð ˜‘Eœ2Ÿ8™8 Q B¨¨a°©c©	°S¸±UÑ';Ñ$;Ó<Ñ<Ð<r8   c                 ó    — d|| z  z
  d|z  z   S )zLcdf = np.exp(-(1.0 - k*x)**(1.0/k))
               logcdf = ...
            r‰   r‡   r¯  s      r6   rœ  zkappa4_gen._logcdf.<locals>.f1Þ  s   € ð ˜1˜Q™3‘Y # a¡%Ñ(Ð(Ð(r8   c                 óh   — d|z  t        j                  | t        j                  |  «      z  «      z  S )zLcdf = (1.0 - h*np.exp(-x))**(1.0/h)
               logcdf = ...
            r‰   )rw   r¦  rP   r·   r¯  s      r6   rë  zkappa4_gen._logcdf.<locals>.f2ä  s,   € ð ˜‘Eœ2Ÿ8™8 Q B¤r§v¡v¨q¨b£z¡MÓ2Ñ2Ð2r8   c                 ó0   — t        j                  |  «       S )zBcdf = np.exp(-np.exp(-x))
               logcdf = ...
            ru  r¯  s      r6   r¡  zkappa4_gen._logcdf.<locals>.f3ê  s   € ô —F‘F˜A˜2“J�;Ðr8   r¥  r©  r³  s	            r6   rã   zkappa4_gen._logcdfÒ  s¢   € Ü—N‘N 1¨¡6¨1°©6Ó2Ü—N‘N 1¨¡6¨1°©6Ó2Ü—N‘N 1¨¡6¨1°©6Ó2Ü—N‘N 1¨¡6¨1°©6Ó2ð4ˆò
	=ò	)ò	3ò	ô ˜8Ø  B¨Ð+Ø˜q !˜9Ü#%§6¡6ô+ð 	+r8   c                 ó>  — t        j                  |dk7  |dk7  «      t        j                  |dk(  |dk7  «      t        j                  |dk7  |dk(  «      t        j                  |dk(  |dk(  «      g}d„ }d„ }d„ }d„ }t        |||||g|||gt         j                  ¬«      S )Nr   c                 ó0   — d|z  dd| |z  z
  |z  |z  z
  z  S r  r‡   ©r}   ró  r  s      r6   r™  zkappa4_gen._ppf.<locals>.f0û  s(   € Ø�q‘5˜# #¨¨A©¡,°Ñ!1°AÑ 5Ñ5Ñ6Ð6r8   c                 óF   — d|z  dt        j                  | «       |z  z
  z  S r  r2  r¼  s      r6   rœ  zkappa4_gen._ppf.<locals>.f1þ  s$   € Ø�q‘5˜#¤"§&¡&¨£) ¨a¡Ñ/Ñ0Ð0r8   c                 ób   — t        j                  | |z   «       t        j                  |«      z   S )z,ppf = -np.log((1.0 - (q**h))/h)
            rÅ  r¼  s      r6   rë  zkappa4_gen._ppf.<locals>.f2  s)   € ô —H‘H˜q !™t˜WÓ%Ð%¬¯©¨q«	Ñ1Ð1r8   c                 óV   — t        j                  t        j                  | «       «       S rN   r2  r¼  s      r6   r¡  zkappa4_gen._ppf.<locals>.f3  s   € Ü—F‘FœBŸF™F 1›I˜:Ó&Ð&Ð&r8   r¥  r©  )	rE   r}   ró  r  rª  r™  rœ  rë  r¡  s	            r6   r~   zkappa4_gen._ppfõ  s¢   € Ü—N‘N 1¨¡6¨1°©6Ó2Ü—N‘N 1¨¡6¨1°©6Ó2Ü—N‘N 1¨¡6¨1°©6Ó2Ü—N‘N 1¨¡6¨1°©6Ó2ð4ˆò
	7ò	1ò	2ò
	'ô ˜8Ø  B¨Ð+Ø˜q !˜9Ü#%§6¡6ô+ð 	+r8   c                 óv   — t        j                  |dk  |dk\  «      |dk  g}d„ }d„ }t        |||g||gd¬«      S )Nr   c                 ó8   — d| z  |z  j                  t        «      S r?  ©Úastyper  rœ  s     r6   r™  z&kappa4_gen._get_stats_info.<locals>.f0  s   € Ø˜‘F˜1‘H×$Ñ$¤SÓ)Ð)r8   c                 ó2   — d|z  j                  t        «      S r?  rÂ  rœ  s     r6   rœ  z&kappa4_gen._get_stats_info.<locals>.f1  s   € Ø˜‘F—?‘?¤3Ó'Ð'r8   rC  r¥  )rP   rƒ  r   )rE   ró  r  rª  r™  rœ  s         r6   Ú_get_stats_infozkappa4_gen._get_stats_info  sK   € ä�N‰N˜1˜q™5 ! q¡&Ó)Ø�‰Eð
ˆò
	*ò	(ô ˜8 b¨" X°°1¨v¸qÔAÐAr8   c                 ó¼   — | j                  ||«      }t        dd«      D �cg c],  }t        j                  ||k  «      rd nt        j                  ‘Œ. }}|d d  S c c}w ©Nr   rC  )rÅ  rý  rP   r£  r  )rE   ró  r  Úmaxrrõ  Úoutputss         r6   r   zkappa4_gen._stats  sU   € Ø×#Ñ# A qÓ)ˆÜAFÀqÈ!ÃÖM¸Aœ2Ÿ6™6 ! d¡(Ô+‘4´·±Ñ7ÐMˆÐMØ‘qˆzÐùò Ns   ¡1Ac                 ó°   — | j                  |d   |d   «      }||k\  rt        j                  S t        j                  | j
                  dd|f|z   ¬«      d   S ©Nr   r   rM  )rÅ  rP   r  r   r¡  Ú_mom_integ1)rE   r?  rG   rÈ  s       r6   Ú_mom1_sczkappa4_gen._mom1_sc!  sR   € Ø×#Ñ# D¨¡G¨T°!©WÓ5ˆØ�Š9Ü—6‘6ˆMÜ�~‰~˜d×.Ñ.°°1¸A¸4À¹9ÔEÀaÑHÐHr8   N)rƒ   r„   r…   r†   rc   rk   r–   rr   rÞ   ru   rã   r~   rÅ  r   rÍ  r‡   r8   r6   r“  r“    sE   „ ñWòp/òò
'òR-ò
$+òL-ò!+òF+ò2Bòó
Ir8   r“  Úkappa4c                   óL   ‡ — e Zd ZdZd„ Zd„ Zd„ Zˆ fd„Zd„ Zd„ Z	d„ Z
d	„ Zˆ xZS )
Ú
kappa3_gena*  Kappa 3 parameter distribution.

    %(before_notes)s

    Notes
    -----
    The probability density function for `kappa3` is:

    .. math::

        f(x, a) = a (a + x^a)^{-(a + 1)/a}

    for :math:`x > 0` and :math:`a > 0`.

    `kappa3` takes ``a`` as a shape parameter for :math:`a`.

    References
    ----------
    P.W. Mielke and E.S. Johnson, "Three-Parameter Kappa Distribution Maximum
    Likelihood and Likelihood Ratio Tests", Methods in Weather Research,
    701-707, (September, 1973),
    :doi:`10.1175/1520-0493(1973)101<0701:TKDMLE>2.3.CO;2`

    B. Kumphon, "Maximum Entropy and Maximum Likelihood Estimation for the
    Three-Parameter Kappa Distribution", Open Journal of Statistics, vol 2,
    415-419 (2012), :doi:`10.4236/ojs.2012.24050`

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r	  rh   rj   s    r6   rk   zkappa3_gen._shape_infoL  r  r8   c                 ó*   — ||||z  z   d|z  dz
  z  z  S r;  r‡   r  s      r6   rr   zkappa3_gen._pdfO  s"   € à�!�a˜‘d‘(˜d 1™f Q™hÑ'Ñ'Ð'r8   c                 ó$   — ||||z  z   d|z  z  z  S r?  r‡   r  s      r6   ru   zkappa3_gen._cdfS  s   € Ø�!�a˜‘d‘(˜d 1™fÑ%Ñ%Ð%r8   c           	      ó  •— t        j                  ||«      \  }}t        ‰| �  ||«      }d}||k  }t	        j
                  t	        j                  d||   z  ||   ||   ||    z  z  «      «       }||kD  }||   |   ||<   |||<   |S )Ng{®Gáz„?r<  )rP   rñ  rA   ry   rw   r  rw  )	rE   rq   r‹   ÚsfÚcutoffrþ  Úsf2Úi2r–  s	           €r6   ry   zkappa3_gen._sfV  sœ   ø€ Ü×"Ñ" 1 aÓ(‰ˆˆ1Ü‰W‰[˜˜AÓˆð
 ˆØ�‰KˆÜ�x‰xœŸ
™
 4¨!¨A©$¡;°°!±°q¸±t¸aÀ¹d¸U±{Ñ0BÓCÓDÐDˆØ�6‰\ˆØ�Q‘%˜‘)ˆˆB‰àˆˆ1‰Øˆ	r8   c                 ó&   — ||| z  dz
  z  d|z  z  S r  r‡   r  s      r6   r~   zkappa3_gen._ppff  s   € Ø�1�q�b‘5˜3‘;‘ 3 q¡5Ñ)Ð)r8   c                 ór   — t        j                  | | «      }t        j                  |«      }||z  d|z  z  S r  r@  )rE   r}   r‹   Úlgr  s        r6   r�   zkappa3_gen._isfi  s6   € Ü�Z‰Z˜˜˜Q˜BÓˆÜ—‘˜“ˆØ�E‘	˜S 1™WÑ%Ð%r8   c                 ó˜   — t        dd«      D �cg c],  }t        j                  ||k  «      rd nt        j                  ‘Œ. }}|d d  S c c}w rÇ  )rý  rP   r£  r  )rE   r‹   rþ  rÉ  s       r6   r   zkappa3_gen._statsn  sB   € Ü>CÀAÀq»kÖJ¸œ2Ÿ6™6 ! a¡%œ=‘4¬b¯f©fÑ4ÐJˆÐJØ‘qˆzÐùò Ks   �1Ac                 ó¬   — t        j                  ||d   k\  «      rt         j                  S t        j                  | j
                  dd|f|z   ¬«      d   S rË  )rP   r£  r  r   r¡  rÌ  )rE   r?  rG   s      r6   rÍ  zkappa3_gen._mom1_scr  sE   € Ü�6‰6�!�t˜A‘w‘,ÔÜ—6‘6ˆMÜ�~‰~˜d×.Ñ.°°1¸A¸4À¹9ÔEÀaÑHÐHr8   )rƒ   r„   r…   r†   rk   rr   ru   ry   r~   r�   r   rÍ  rÐ  rÑ  s   @r6   rÐ  rÐ  +  s3   ø„ ñò@Eò(ò&ôò *ò&ò
öIr8   rÐ  Úkappa3c                   óB   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zy)Ú	moyal_genaÄ  A Moyal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `moyal` is:

    .. math::

        f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi}

    for a real number :math:`x`.

    %(after_notes)s

    This distribution has utility in high-energy physics and radiation
    detection. It describes the energy loss of a charged relativistic
    particle due to ionization of the medium [1]_. It also provides an
    approximation for the Landau distribution. For an in depth description
    see [2]_. For additional description, see [3]_.

    References
    ----------
    .. [1] J.E. Moyal, "XXX. Theory of ionization fluctuations",
           The London, Edinburgh, and Dublin Philosophical Magazine
           and Journal of Science, vol 46, 263-280, (1955).
           :doi:`10.1080/14786440308521076` (gated)
    .. [2] G. Cordeiro et al., "The beta Moyal: a useful skew distribution",
           International Journal of Research and Reviews in Applied Sciences,
           vol 10, 171-192, (2012).
           http://www.arpapress.com/Volumes/Vol10Issue2/IJRRAS_10_2_02.pdf
    .. [3] C. Walck, "Handbook on Statistical Distributions for
           Experimentalists; International Report SUF-PFY/96-01", Chapter 26,
           University of Stockholm: Stockholm, Sweden, (2007).
           http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf

    .. versionadded:: 1.1.0

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zmoyal_gen._shape_info¦  r¦   r8   Nc                 ó`   — t         j                  dd||¬«      }t        j                  |«       S )Nr”   rU   )r‹   r/   r×   rØ   )rØ  rÙ  rP   rð   )rE   r×   rØ   rÚ  s       r6   rÙ   zmoyal_gen._rvs©  s.   € Ü�Y‰Y˜ A¨DØ$0ð ó 2ˆä—‘�r“
ˆ{Ðr8   c                 ó®   — t        j                  d|t        j                  | «      z   z  «      t        j                  dt         j                  z  «      z  S ©Nr$  rU   )rP   r·   rÿ   rñ   r©   s     r6   rr   zmoyal_gen._pdf®  s:   € Ü�v‰v�d˜a¤"§&¡&¨!¨£*™nÑ-Ó.´·±¸¼2¿5¹5¹Ó1AÑAÐAr8   c                 ó„   — t        j                  t        j                  d|z  «      t        j                  d«      z  «      S rä  )rw   rÌ  rP   r·   rÿ   r©   s     r6   ru   zmoyal_gen._cdf±  s+   € Ü�w‰w”r—v‘v˜d Q™hÓ'¬"¯'©'°!«*Ñ4Ó5Ð5r8   c                 ó„   — t        j                  t        j                  d|z  «      t        j                  d«      z  «      S rä  )rw   r   rP   r·   rÿ   r©   s     r6   ry   zmoyal_gen._sf´  s+   € Ü�v‰v”b—f‘f˜T A™XÓ&¬¯©°«Ñ3Ó4Ð4r8   c                 ó`   — t        j                  dt        j                  |«      dz  z  «       S r  )rP   rð   rw   Úerfcinvr©   s     r6   r~   zmoyal_gen._ppf·  s&   € Ü—‘�qœ2Ÿ:™: a›=¨!Ñ+Ñ+Ó,Ð,Ð,r8   c                 ó  — t        j                  d«      t         j                  z   }t         j                  dz  dz  }dt        j                  d«      z  t        j                  d«      z  t         j                  dz  z  }d}||||fS )NrU   é   r†  rJ  )rP   rð   Úeuler_gammarñ   rÿ   rw   r�  rB  s        r6   r   zmoyal_gen._statsº  sg   € Ü�V‰V�A‹YœŸ™Ñ'ˆÜ�e‰e�Q‰h˜‰lˆØ”"—'‘'˜!“*‰_œrŸw™w q›zÑ)¬B¯E©E°1©HÑ4ˆØˆØ�3˜˜BˆÐr8   c                 ó¼  — |dk(  r&t        j                  d«      t         j                  z   S |dk(  r@t         j                  dz  dz  t        j                  d«      t         j                  z   dz  z   S |dk(  r†dt         j                  dz  z  t        j                  d«      t         j                  z   z  }t        j                  d«      t         j                  z   dz  }dt	        j
                  d«      z  }||z   |z   S |dk(  rÌd	t	        j
                  d«      z  t        j                  d«      t         j                  z   z  }dt         j                  dz  z  t        j                  d«      t         j                  z   dz  z  }t        j                  d«      t         j                  z   d
z  }dt         j                  d
z  z  d
z  }||z   |z   |z   S | j                  |«      S )Nr‰   rU   r¶   rD  rÊ  r†  r  rJ  é8   r$  rÉ  )rP   rð   rë  rñ   rw   r�  rÍ  )rE   rb   Útmp1r{  Útmp3Útmp4s         r6   r  zmoyal_gen._munpÁ  sn  € Ø�Š8Ü—6‘6˜!“9œrŸ~™~Ñ-Ð-Ø�#ŠXÜ—5‘5˜!‘8˜a‘<¤2§6¡6¨!£9¬r¯~©~Ñ#=ÀÑ"AÑAÐAØ�#ŠXØœŸ™ ™‘>¤R§V¡V¨A£Y¬r¯~©~Ñ%=Ñ>ˆDÜ—F‘F˜1“IœbŸn™nÑ,¨qÑ0ˆDØœŸ™ ›
‘?ˆDØ˜$‘; Ñ%Ð%Ø�#ŠXØœBŸG™G A›JÑ&¬"¯&©&°«)´b·n±nÑ*DÑEˆDØ”r—u‘u˜a‘x‘<¤2§6¡6¨!£9¬r¯~©~Ñ#=ÀÑ"AÑAˆDÜ—F‘F˜1“I¤§¡Ñ.°Ñ2ˆDØ”r—u‘u˜a‘x‘< !Ñ#ˆDØ˜$‘; Ñ%¨Ñ,Ð,ð —=‘= Ó#Ð#r8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   ru   ry   r~   r   r  r‡   r8   r6   rà  rà  {  s1   „ ñ)òTóò
Bò6ò5ò-òó$r8   rà  Úmoyalc                   ó\   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zdd„Zdd„Zy)Únakagami_gena`  A Nakagami continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `nakagami` is:

    .. math::

        f(x, \nu) = \frac{2 \nu^\nu}{\Gamma(\nu)} x^{2\nu-1} \exp(-\nu x^2)

    for :math:`x >= 0`, :math:`\nu > 0`. The distribution was introduced in
    [2]_, see also [1]_ for further information.

    `nakagami` takes ``nu`` as a shape parameter for :math:`\nu`.

    %(after_notes)s

    References
    ----------
    .. [1] "Nakagami distribution", Wikipedia
           https://en.wikipedia.org/wiki/Nakagami_distribution
    .. [2] M. Nakagami, "The m-distribution - A general formula of intensity
           distribution of rapid fading", Statistical methods in radio wave
           propagation, Pergamon Press, 1960, 3-36.
           :doi:`10.1016/B978-0-08-009306-2.50005-4`

    %(example)s

    c                 ó   — |dkD  S r2  r‡   )rE   Únus     r6   rc   znakagami_gen._argcheckú  r¿  r8   c                 ó@   — t        dddt        j                  fd«      gS )Nrõ  Fr   r
  rh   rj   s    r6   rk   znakagami_gen._shape_infoý  r²  r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  ©rE   rq   rõ  s      r6   rr   znakagami_gen._pdf   rŸ  r8   c                 óÒ   — t        j                  d«      t        j                  ||«      z   t        j                  |«      z
  t        j                  d|z  dz
  |«      z   ||dz  z  z
  S r‹  )rP   rð   rw   rx  rÆ  rø  s      r6   rÞ   znakagami_gen._logpdf  s\   € ô —‘�q“	œBŸH™H R¨Ó,Ñ,¬r¯z©z¸"«~Ñ=Ü—‘˜˜2™ ™ 1Ó%ñ&Ø(*¨1¨a©4©ñ0ð 	1r8   c                 ó:   — t        j                  |||z  |z  «      S rN   r¼  rø  s      r6   ru   znakagami_gen._cdf	  s   € Ü�{‰{˜2˜r !™t A™vÓ&Ð&r8   c                 ó`   — t        j                  d|z  t        j                  ||«      z  «      S r  rÃ  )rE   r}   rõ  s      r6   r~   znakagami_gen._ppf  s%   € Ü�w‰w�s˜2‘vœbŸn™n¨R°Ó3Ñ3Ó4Ð4r8   c                 ó:   — t        j                  |||z  |z  «      S rN   r¿  rø  s      r6   ry   znakagami_gen._sf  s   € Ü�|‰|˜B  1¡ Q¡Ó'Ð'r8   c                 ó`   — t        j                  d|z  t        j                  ||«      z  «      S r^   rÇ  )rE   rô  rõ  s      r6   r�   znakagami_gen._isf  s%   € Ü�w‰w�q˜‘tœbŸo™o¨b°!Ó4Ñ4Ó5Ð5r8   c                 ó(  — t        j                  |d«      t        j                  |«      z  }d||z  z
  }|dd|z  |z  z
  z  dz  |z  t        j                  |d«      z  }d|dz  z  |z  d|z  d	z
  |d	z  z  z   d	|z  z
  dz   }|||dz  z  z  }||||fS )
Nr”   r‰   r   r$  r¶   rÊ  éúÿÿÿr.  rU   )rw   rË  rP   rÿ   rÌ  )rE   rõ  rC  rD  rE  rF  s         r6   r   znakagami_gen._stats  s°   € Ü�W‰W�R˜ÓœbŸg™g b›kÑ)ˆØ�"�R‘%‰iˆØ�1�q˜‘t˜C‘x‘<Ñ  3Ñ&¨Ñ+¬b¯h©h°s¸CÓ.@Ñ@ˆØ��A‘‰X�b‰[˜A˜b™D ™F B¨¡E™>Ñ)¨!¨B©$Ñ.°Ñ2ˆØ
ˆb��c‘‰kÑˆØ�3˜˜BˆÐr8   c                 óÄ  — t        j                  |«      }t        j                  |«      }t        j                  |«      }||dz
  t        j
                  |«      z  z
  }dt        j                  |«      z  t        j                  d«      z
  }||z   |z   }t        j                  j                  «       }|dkD  }||   |z   dd||   z  z  z
  ||<   |j                  |«      d   S )Nr”   r$  rU   g     jè@r   rÀ  r‡   )rP   r·  rã  rw   rÆ  rÐ  rð   rò  r  rò   rN  )	rE   rõ  r·  rm  rn  rC  ró  Únorm_entropyrþ  s	            r6   rò   znakagami_gen._entropy  sÅ   € Ü—‘˜“ˆä�]‰]˜2ÓˆÜ�J‰J�r‹NˆØ�"�s‘(œbŸj™j¨›nÑ,Ñ,ˆØ”2—6‘6˜"“:Ñ¤§¡ q£	Ñ)ˆØ�‰E�A‰Iˆä—z‘z×*Ñ*Ó,ˆð �‰Hˆà�‰t�lÑ" Q¨¨2¨a©5©¡\Ñ1ˆˆ!‰Ø�y‰y˜Ó Ñ#Ð#r8   Nc                 óT   — t        j                  |j                  ||¬«      |z  «      S r4  )rP   rÿ   r%  )rE   rõ  r×   rØ   s       r6   rÙ   znakagami_gen._rvs.  s&   € ä�w‰w�|×2Ñ2°2¸DÐ2ÓAÀBÑFÓGÐGr8   c                 ó  — t        |t        «      r|j                  «       }|€d| j                  z  }t	        j
                  |«      }t	        j                  t	        j                  ||z
  dz  «      t        |«      z  «      }|||fz   S )N)r‰   rU   )	r?   r*   r“  ÚnumargsrP   r‡  rÿ   r¥  r¤  )rE   rF   rG   r.   r/   s        r6   r•  znakagami_gen._fitstart2  sq   € Ü�dœLÔ)Ø—>‘>Ó#ˆDØˆ<Ø˜DŸL™LÑ(ˆDô �f‰f�T‹lˆÜ—‘œŸ™  s¡
¨Q™Ó/´#°d³)Ñ;Ó<ˆØ�s˜E�lÑ"Ð"r8   r  rN   )rƒ   r„   r…   r†   rc   rk   rr   rÞ   ru   r~   ry   r�   r   rò   rÙ   r•  r‡   r8   r6   ró  ró  Ú  sE   „ ñò>òFò+ò1ò'ò5ò(ò6òò$ó"Hô	#r8   ró  Únakagamic                 ó6  — |dz  dz
  }t        j                  | «      t        j                  |«      }}t        j                  |dz  | |z  «      d||z
  dz  z  z
  }t        j                  |||z  «      dz  }t        |dkD  ||fd„ t         j                   ¬«      S )Nr¶   r‰   r”   rU   r   c                 ó2   — | t        j                  |«      z   S rN   r2  )rõ  r  s     r6   rç  z_ncx2_log_pdf.<locals>.<lambda>N  s   € �qœ2Ÿ6™6 !›9‘}€ r8   rù  )rP   rÿ   rw   rx  Úiver   ri   )rq   r±  rQ  Údf2rü  Únsrõ  Úcorrs           r6   Ú_ncx2_log_pdfr	  B  s”   € ð ˆS‰&�3‰,€CÜ�W‰W�Q‹ZœŸ™ ›ˆ€BÜ
�(‰(�3�s‘7˜A˜b™DÓ
! C¨¨b©°1©Ñ$4Ñ
4€CÜ�6‰6�#�r˜"‘uÓ Ñ#€DäØˆq‰Ø	ˆdˆÙ
$Ü—6‘6�'ô	ð r8   c                   óN   — e Zd ZdZd„ Zd„ Zdd„Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Zd„ Zy)Úncx2_gena  A non-central chi-squared continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `ncx2` is:

    .. math::

        f(x, k, \lambda) = \frac{1}{2} \exp(-(\lambda+x)/2)
            (x/\lambda)^{(k-2)/4}  I_{(k-2)/2}(\sqrt{\lambda x})

    for :math:`x >= 0`, :math:`k > 0` and :math:`\lambda \ge 0`.
    :math:`k` specifies the degrees of freedom (denoted ``df`` in the
    implementation) and :math:`\lambda` is the non-centrality parameter
    (denoted ``nc`` in the implementation). :math:`I_\nu` denotes the
    modified Bessel function of first order of degree :math:`\nu`
    (`scipy.special.iv`).

    `ncx2` takes ``df`` and ``nc`` as shape parameters.

    This distribution uses routines from the Boost Math C++ library for
    the computation of the ``pdf``, ``cdf``, ``ppf``, ``sf`` and ``isf``
    methods. [1]_

    %(after_notes)s

    References
    ----------
    .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

    %(example)s

    c                 óD   — |dkD  t        j                  |«      z  |dk\  z  S r2  r™  ©rE   r±  rQ  s      r6   rc   zncx2_gen._argcheckv  s"   € Ø�Q‘œ"Ÿ+™+ b›/Ñ)¨R°1©WÑ5Ð5r8   c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS )Nr±  Fr   r
  rQ  rg   rh   ©rE   ÚidfÚincs      r6   rk   zncx2_gen._shape_infoy  s<   € Ü˜˜u q¬"¯&©& k°>ÓBˆÜ˜˜u q¬"¯&©& k°=ÓAˆØ�SˆzÐr8   Nc                 ó(   — |j                  |||«      S rN   )Únoncentral_chisquare)rE   r±  rQ  r×   rØ   s        r6   rÙ   zncx2_gen._rvs~  s   € Ø×0Ñ0°°R¸Ó>Ð>r8   c                 ór   — t        j                  |t        ¬«      |dk7  z  }t        ||||ft        d„ ¬«      S )Nr  r   c                 ó.   — t         j                  | |«      S rN   )r´  rÞ   ©rq   r±  Ú_s      r6   rç  z"ncx2_gen._logpdf.<locals>.<lambda>„  s   € ¬d¯l©l¸1¸bÓ.A€ r8   rŒ  )rP   Ú	ones_likeÚboolr   r	  ©rE   rq   r±  rQ  Úconds        r6   rÞ   zncx2_gen._logpdf�  s9   € Ü�|‰|˜A¤TÔ*¨b°A©gÑ6ˆÜ˜$  B¨ ¬}ÙAôCð 	Cr8   c                 óÞ   — t        j                  |t        ¬«      |dk7  z  }t        j                  d¬«      5  t	        ||||ft
        j                  d„ ¬«      cd d d «       S # 1 sw Y   y xY w)Nr  r   r9  rq  c                 ó.   — t         j                  | |«      S rN   )r´  rr   r	  s      r6   rç  zncx2_gen._pdf.<locals>.<lambda>Š  ó   € ´$·)±)¸A¸rÓ2B€ r8   rŒ  )rP   r	  r	  r<  r   rn   Ú	_ncx2_pdfr	  s        r6   rr   zncx2_gen._pdf†  ó\   € Ü�|‰|˜A¤TÔ*¨b°A©gÑ6ˆÜ�[‰[˜hÔ'ñ 	DÜ˜d Q¨¨B K´3·=±=Ù!BôD÷	D÷ 	Dò 	Dúó   ¸!A#Á#A,c                 óÞ   — t        j                  |t        ¬«      |dk7  z  }t        j                  d¬«      5  t	        ||||ft
        j                  d„ ¬«      cd d d «       S # 1 sw Y   y xY w)Nr  r   r9  rq  c                 ó.   — t         j                  | |«      S rN   )r´  ru   r	  s      r6   rç  zncx2_gen._cdf.<locals>.<lambda>�  r!	  r8   rŒ  )rP   r	  r	  r<  r   rn   Ú	_ncx2_cdfr	  s        r6   ru   zncx2_gen._cdfŒ  r#	  r$	  c                 óÞ   — t        j                  |t        ¬«      |dk7  z  }t        j                  d¬«      5  t	        ||||ft
        j                  d„ ¬«      cd d d «       S # 1 sw Y   y xY w)Nr  r   r9  rq  c                 ó.   — t         j                  | |«      S rN   )r´  r~   r	  s      r6   rç  zncx2_gen._ppf.<locals>.<lambda>–  r!	  r8   rŒ  )rP   r	  r	  r<  r   rn   Ú	_ncx2_ppf)rE   r}   r±  rQ  r	  s        r6   r~   zncx2_gen._ppf’  r#	  r$	  c                 óÞ   — t        j                  |t        ¬«      |dk7  z  }t        j                  d¬«      5  t	        ||||ft
        j                  d„ ¬«      cd d d «       S # 1 sw Y   y xY w)Nr  r   r9  rq  c                 ó.   — t         j                  | |«      S rN   )r´  ry   r	  s      r6   rç  zncx2_gen._sf.<locals>.<lambda>œ  s   € ´$·(±(¸1¸b³/€ r8   rŒ  )rP   r	  r	  r<  r   rn   Ú_ncx2_sfr	  s        r6   ry   zncx2_gen._sf˜  s\   € Ü�|‰|˜A¤TÔ*¨b°A©gÑ6ˆÜ�[‰[˜hÔ'ñ 	CÜ˜d Q¨¨B K´3·<±<Ù!AôC÷	C÷ 	Cò 	Cúr$	  c                 óÞ   — t        j                  |t        ¬«      |dk7  z  }t        j                  d¬«      5  t	        ||||ft
        j                  d„ ¬«      cd d d «       S # 1 sw Y   y xY w)Nr  r   r9  rq  c                 ó.   — t         j                  | |«      S rN   )r´  r�   r	  s      r6   rç  zncx2_gen._isf.<locals>.<lambda>¢  r!	  r8   rŒ  )rP   r	  r	  r<  r   rn   Ú	_ncx2_isfr	  s        r6   r�   zncx2_gen._isfž  r#	  r$	  c                 óð   — ||z   }d„ }d |||d«      z  }t        j                  d«       |||d«      z  t        j                   |||d«      dz  «      z  }d |||d«      z   |||d«      dz  z  }||||fS )Nc                 ó   — | ||z  z   S rN   r‡   )r  r¹  r  s      r6   Ú	k_plus_clz"ncx2_gen._stats.<locals>.k_plus_cl¦  s   € Ø�q˜‘s‘7ˆNr8   r¶   r	  r†  rí  rJ  rU   r‡  )rE   r±  rQ  Ú
_ncx2_meanr3	  Ú_ncx2_varianceÚ_ncx2_skewnessÚ_ncx2_kurtosis_excesss           r6   r   zncx2_gen._stats¤  sŸ   € Ø˜"‘Wˆ
ò	à¡	¨"¨b°#Ó 6Ñ6ˆÜŸ'™' #›,©°2°r¸1Ó)=Ñ=ÜŸ'™'¡)¨B°°CÓ"8¸!Ñ";Ó<ñ=ˆà!%©	°"°b¸#Ó(>Ñ!>Ù!*¨2¨r°3Ó!7¸Ñ!:ñ";Ðð ØØØ!ð	
ð 	
r8   r  )rƒ   r„   r…   r†   rc   rk   rÙ   rÞ   rr   ru   r~   ry   r�   r   r‡   r8   r6   r	  r	  R  s@   „ ñ"òF6òó
?òCò
DòDòDòCòDó
r8   r	  Úncx2c                   óJ   — e Zd ZdZd„ Zd„ Zdd„Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zdd„Zy)Úncf_gena2  A non-central F distribution continuous random variable.

    %(before_notes)s

    See Also
    --------
    scipy.stats.f : Fisher distribution

    Notes
    -----
    The probability density function for `ncf` is:

    .. math::

        f(x, n_1, n_2, \lambda) =
            \exp\left(\frac{\lambda}{2} +
                      \lambda n_1 \frac{x}{2(n_1 x + n_2)}
                \right)
            n_1^{n_1/2} n_2^{n_2/2} x^{n_1/2 - 1} \\
            (n_2 + n_1 x)^{-(n_1 + n_2)/2}
            \gamma(n_1/2) \gamma(1 + n_2/2) \\
            \frac{L^{\frac{n_1}{2}-1}_{n_2/2}
                \left(-\lambda n_1 \frac{x}{2(n_1 x + n_2)}\right)}
            {B(n_1/2, n_2/2)
                \gamma\left(\frac{n_1 + n_2}{2}\right)}

    for :math:`n_1, n_2 > 0`, :math:`\lambda \ge 0`.  Here :math:`n_1` is the
    degrees of freedom in the numerator, :math:`n_2` the degrees of freedom in
    the denominator, :math:`\lambda` the non-centrality parameter,
    :math:`\gamma` is the logarithm of the Gamma function, :math:`L_n^k` is a
    generalized Laguerre polynomial and :math:`B` is the beta function.

    `ncf` takes ``dfn``, ``dfd`` and ``nc`` as shape parameters. If ``nc=0``,
    the distribution becomes equivalent to the Fisher distribution.

    This distribution uses routines from the Boost Math C++ library for
    the computation of the ``pdf``, ``cdf``, ``ppf``, ``stats``, ``sf`` and
    ``isf`` methods. [1]_

    %(after_notes)s

    References
    ----------
    .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

    %(example)s

    c                 ó$   — |dkD  |dkD  z  |dk\  z  S r2  r‡   )rE   rø  rù  rQ  s       r6   rc   zncf_gen._argchecké  s   € Ø�a‘˜C !™GÑ$¨¨a©Ñ0Ð0r8   c                 óÀ   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }t        dddt        j                  fd«      }|||gS )Nrø  Fr   r
  rù  rQ  rg   rh   )rE   Úidf1Úidf2r	  s       r6   rk   zncf_gen._shape_infoì  sW   € Ü˜% ¨¬B¯F©F¨°^ÓDˆÜ˜% ¨¬B¯F©F¨°^ÓDˆÜ˜˜u q¬"¯&©& k°=ÓAˆØ�d˜CÐ Ð r8   Nc                 ó*   — |j                  ||||«      S rN   )Únoncentral_f)rE   rø  rù  rQ  r×   rØ   s         r6   rÙ   zncf_gen._rvsò  s   € Ø×(Ñ(¨¨c°2°tÓ<Ð<r8   c                 ó2   — t        j                  ||||«      S rN   )rn   Ú_ncf_pdf©rE   rq   rø  rù  rQ  s        r6   rr   zncf_gen._pdfõ  s   € Ü�|‰|˜A˜s C¨Ó,Ð,r8   c                 ó2   — t        j                  ||||«      S rN   )rw   ÚncfdtrrC	  s        r6   ru   zncf_gen._cdfø  s   € Ü�y‰y˜˜c 2 qÓ)Ð)r8   c                 óŠ   — t        j                  d¬«      5  t        j                  ||||«      cd d d «       S # 1 sw Y   y xY wrp  )rP   r<  rw   Úncfdtri)rE   r}   rø  rù  rQ  s        r6   r~   zncf_gen._ppfû  s5   € Ü�[‰[˜hÔ'ñ 	/Ü—:‘:˜c 3¨¨AÓ.÷	/÷ 	/ò 	/úó	   —9¹Ac                 ó2   — t        j                  ||||«      S rN   )rn   Ú_ncf_sfrC	  s        r6   ry   zncf_gen._sfÿ  s   € Ü�{‰{˜1˜c 3¨Ó+Ð+r8   c                 óŠ   — t        j                  d¬«      5  t        j                  ||||«      cd d d «       S # 1 sw Y   y xY wrp  )rP   r<  rn   Ú_ncf_isfrC	  s        r6   r�   zncf_gen._isf  s5   € Ü�[‰[˜hÔ'ñ 	1Ü—<‘<  3¨¨RÓ0÷	1÷ 	1ò 	1úrH	  c                 óä   — t        j                  |||«      }t        j                  |||«      }d|v rt        j                  |||«      nd }d|v rt        j                  |||«      dz
  nd }||||fS )Nr  r  r†  )rn   Ú	_ncf_meanÚ_ncf_varianceÚ_ncf_skewnessÚ_ncf_kurtosis_excess)	rE   rø  rù  rQ  r  rC  rD  rE  rF  s	            r6   r   zncf_gen._stats  s‚   € Ü�]‰]˜3  RÓ(ˆÜ×Ñ  S¨"Ó-ˆØ03°w±ŒS×Ñ˜s C¨Ô,ÀDˆà!$¨¡ô ×%Ñ%Ø��bóØòØ59ð 	ð �3˜˜BˆÐr8   r  r  ©rƒ   r„   r…   r†   rc   rk   rÙ   rr   ru   r~   ry   r�   r   r‡   r8   r6   r:	  r:	  ¸  s5   „ ñ/ò`1ò!ó=ò-ò*ò/ò,ò1ôr8   r:	  Úncfc                   óN   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zd„ Zd„ Zy)Út_gena‹  A Student's t continuous random variable.

    For the noncentral t distribution, see `nct`.

    %(before_notes)s

    See Also
    --------
    nct

    Notes
    -----
    The probability density function for `t` is:

    .. math::

        f(x, \nu) = \frac{\Gamma((\nu+1)/2)}
                        {\sqrt{\pi \nu} \Gamma(\nu/2)}
                    (1+x^2/\nu)^{-(\nu+1)/2}

    where :math:`x` is a real number and the degrees of freedom parameter
    :math:`\nu` (denoted ``df`` in the implementation) satisfies
    :math:`\nu > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r°  rh   rj   s    r6   rk   zt_gen._shape_info:  r²  r8   Nc                 ó(   — |j                  ||¬«      S r4  )Ú
standard_trµ  s       r6   rÙ   z
t_gen._rvs=  s   € Ø×&Ñ& r°Ð&Ó5Ð5r8   c                 óP   ‡ — t        |t        j                  k(  ||fd„ ˆ fd„¬«      S )Nc                 ó,   — t         j                  | «      S rN   )r  rr   ©rq   r±  s     r6   rç  zt_gen._pdf.<locals>.<lambda>C  s   € œDŸI™I a›L€ r8   c                 óN   •— t        j                  ‰j                  | |«      «      S rN   rÝ  )rq   r±  rE   s     €r6   rç  zt_gen._pdf.<locals>.<lambda>D  s   ø€ Ü—‘�t—|‘| A rÓ*Ó+ð r8   rŒ  r  r·  s   `  r6   rr   z
t_gen._pdf@  s)   ø€ ÜØ”"—&‘&‰L˜1˜b˜'Ù(óô
ð 	
r8   c                 óR   — d„ }d„ }t        |t        j                  k(  ||f||¬«      S )Nc                 ó  — t        j                  t        j                  d|z  d«      «      dt        j                  |«      t        j                  t         j                  «      z   z  z
  |dz   dz  t        j
                  | | z  |z  «      z  z
  S rK  )rP   rð   rw   rË  rñ   r¦  r[	  s     r6   Út_logpdfzt_gen._logpdf.<locals>.t_logpdfK  sn   € Ü—F‘Fœ2Ÿ7™7 3¨¡8¨SÓ1Ó2ØœRŸV™V B›Z¬"¯&©&´·±«-Ñ7Ñ8ñ9à˜A‘v˜q‘j¤§¡¨!¨a©%°©(Ó!3Ñ3ñ4ð 5r8   c                 ó,   — t         j                  | «      S rN   )r  rÞ   r[	  s     r6   Únorm_logpdfz"t_gen._logpdf.<locals>.norm_logpdfP  s   € Ü—<‘< “?Ð"r8   rŒ  r  )rE   rq   r±  r_	  ra	  s        r6   rÞ   zt_gen._logpdfI  s+   € ò	5ò
	#ô ˜"¤§¡™,¨¨B¨	°[ÀXÔNÐNr8   c                 ó.   — t        j                  ||«      S rN   ©rw   Ústdtrr·  s      r6   ru   z
t_gen._cdfU  rã  r8   c                 ó0   — t        j                  || «      S rN   rc	  r·  s      r6   ry   z	t_gen._sfX  s   € Ü�x‰x˜˜Q˜BÓÐr8   c                 ó.   — t        j                  ||«      S rN   ©rw   ÚstdtritrÅ  s      r6   r~   z
t_gen._ppf[  s   € Ü�z‰z˜"˜aÓ Ð r8   c                 ó0   — t        j                  ||«       S rN   rg	  rÅ  s      r6   r�   z
t_gen._isf^  s   € Ü—
‘
˜2˜qÓ!Ð!Ð!r8   c                 ó  — t        j                  |«      }t        j                  |dkD  dt         j                  «      }|dkD  |dk  z  |dkD  t        j                  |«      z  |f}d„ d„ d„ f}t        |||ft         j                  «      }t        j                  |dkD  dt         j                  «      }|dkD  |dk  z  |dkD  t        j                  |«      z  |f}d	„ d
„ d„ f}t        |||ft         j                  «      }||||fS )Nr   rˆ   rU   c                 ó^   — t        j                  t         j                  | j                  «      S rN   ©rP   Úbroadcast_tori   r·  rÑ  s    r6   rç  zt_gen._stats.<locals>.<lambda>j  ó   € ¤§¡´·±¸¿¹Ó!B€ r8   c                 ó   — | | dz
  z  S r>  r‡   rÑ  s    r6   rç  zt_gen._stats.<locals>.<lambda>k  s   €   r¨#¡v¡€ r8   c                 óB   — t        j                  d| j                  «      S r^   ©rP   rm	  r·  rÑ  s    r6   rç  zt_gen._stats.<locals>.<lambda>l  ó   € ¤§¡°°B·H±HÓ!=€ r8   r†  r$  c                 ó^   — t        j                  t         j                  | j                  «      S rN   rl	  rÑ  s    r6   rç  zt_gen._stats.<locals>.<lambda>t  rn	  r8   c                 ó   — d| dz
  z  S )Nr�  rJ  r‡   rÑ  s    r6   rç  zt_gen._stats.<locals>.<lambda>u  s   €  ¨¨3©¡€ r8   c                 óB   — t        j                  d| j                  «      S r2  rq	  rÑ  s    r6   rç  zt_gen._stats.<locals>.<lambda>v  rr	  r8   )rP   ÚisposinfrO  ri   rü   r   r  )	rE   r±  Úinfinite_dfrC  rª  Ú
choicelistrD  rE  rF  s	            r6   r   zt_gen._statsa  s  € ä—k‘k "“oˆä�X‰X�b˜1‘f˜c¤2§6¡6Ó*ˆà˜!‘V  a¡Ñ(Ø˜!‘VœrŸ{™{¨2›Ñ.Øð!ˆñ CÙ.Ù=ð?ˆ
ô ˜( J°°´r·v±vÓ>ˆä�X‰X�b˜1‘f˜c¤2§6¡6Ó*ˆà˜!‘V  a¡Ñ(Ø˜!‘VœrŸ{™{¨2›Ñ.Øð!ˆñ CÙ/Ù=ð?ˆ
ô ˜ :°¨u´b·f±fÓ=ˆà�3˜˜BˆÐr8   c                 ó†   — |t         j                  k(  rt        j                  «       S d„ }d„ }t	        |dk\  |f||¬«      }|S )Nc                 óø   — | dz  }| dz   dz  }|t        j                  |«      t        j                  |«      z
  z  t        j                  t        j                  | «      t        j
                  |d«      z  «      z   S rE  )rw   rÐ  rP   rð   rÿ   rn  )r±  ÚhalfÚhalf1s      r6   r¯  zt_gen._entropy.<locals>.regular  se   € Ø�a‘4ˆDØ˜!‘V˜Q‘JˆEØœ2Ÿ:™: eÓ,¬r¯z©z¸$Ó/?Ñ?Ñ@Ü—f‘fœRŸW™W R›[¬¯©°°sÓ);Ñ;Ó<ñ=ð >r8   c                 ó”   — t         j                  «       d| z  z   | dz  dz  z   | dz  dz  z
  | dz  dz  z
  d| d	z  z  z   | d
z  dz  z   }|S )Nr   r³  r$  r´  r…  rµ  r.  g333333Ó?rÁ  rÃ  )r  rò   )r±  ró  s     r6   rp  z"t_gen._entropy.<locals>.asymptotic…  sg   € ô —‘“ 1 R¡4Ñ'¨2¨s©7°A©+Ñ5¸¸S¹À!¹ÑCØ˜‘G˜Q‘;ñØ!% r¨3¡w¡ñ0Ø35°s±7¸A±+ñ>ˆAàˆHr8   éd   rŒ  )rP   ri   r  rò   r   )rE   r±  r¯  rp  ró  s        r6   rò   zt_gen._entropy{  s@   € Ø”—‘Š<Ü—=‘=“?Ð"ò	>ò	ô �r˜S‘y 2 &¨J¸7ÔCˆØˆr8   r  rØ  r‡   r8   r6   rU	  rU	    s;   „ ñò<Fó6ò
ò
Oòò ò!ò"òó4r8   rU	  rÏ  c                   óJ   — e Zd ZdZd„ Zd„ Zdd„Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zdd„Zy)Únct_gena™  A non-central Student's t continuous random variable.

    %(before_notes)s

    Notes
    -----
    If :math:`Y` is a standard normal random variable and :math:`V` is
    an independent chi-square random variable (`chi2`) with :math:`k` degrees
    of freedom, then

    .. math::

        X = \frac{Y + c}{\sqrt{V/k}}

    has a non-central Student's t distribution on the real line.
    The degrees of freedom parameter :math:`k` (denoted ``df`` in the
    implementation) satisfies :math:`k > 0` and the noncentrality parameter
    :math:`c` (denoted ``nc`` in the implementation) is a real number.

    This distribution uses routines from the Boost Math C++ library for
    the computation of the ``pdf``, ``cdf``, ``ppf``, ``sf`` and ``isf``
    methods. [1]_

    %(after_notes)s

    References
    ----------
    .. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

    %(example)s

    c                 ó   — |dkD  ||k(  z  S r2  r‡   r	  s      r6   rc   znct_gen._argcheckµ  s   € Ø�Q‘˜2 ™8Ñ$Ð$r8   c                 ó    — t        dddt        j                  fd«      }t        ddt        j                   t        j                  fd«      }||gS )Nr±  Fr   r
  rQ  rh   r	  s      r6   rk   znct_gen._shape_info¸  sC   € Ü˜˜u q¬"¯&©& k°>ÓBˆÜ˜˜u¬¯© w´·±Ð&7¸ÓHˆØ�SˆzÐr8   Nc                 ó¾   — t         j                  |||¬«      }t        j                  |||¬«      }|t        j                  |«      z  t        j                  |«      z  S )Nrê  r×  )r  rÙ  r´  rP   rÿ   )rE   r±  rQ  r×   rØ   rb   rß  s          r6   rÙ   znct_gen._rvs½  sI   € Ü�H‰H˜ $°\ˆHÓBˆÜ�X‰X�b˜t°,ˆXÓ?ˆØ”2—7‘7˜2“;‰¤§¡¨£Ñ,Ð,r8   c                 ó0   — t        j                  |||«      S rN   )rn   Ú_nct_pdf©rE   rq   r±  rQ  s       r6   rr   znct_gen._pdfÂ  s   € Ü�|‰|˜A˜r 2Ó&Ð&r8   c                 ó0   — t        j                  |||«      S rN   )rw   Únctdtrr†	  s       r6   ru   znct_gen._cdfÅ  s   € Ü�y‰y˜˜R Ó#Ð#r8   c                 óˆ   — t        j                  d¬«      5  t        j                  |||«      cd d d «       S # 1 sw Y   y xY wrp  )rP   r<  rn   Ú_nct_ppf)rE   r}   r±  rQ  s       r6   r~   znct_gen._ppfÈ  ó3   € Ü�[‰[˜hÔ'ñ 	+Ü—<‘<  2 rÓ*÷	+÷ 	+ò 	+úru  c                 ó²   — t        j                  d¬«      5  t        j                  t        j                  |||«      dd«      cd d d «       S # 1 sw Y   y xY w)Nr9  rq  r   r   )rP   r<  Úcliprn   Ú_nct_sfr†	  s       r6   ry   znct_gen._sfÌ  s@   € Ü�[‰[˜hÔ'ñ 	9Ü—7‘7œ3Ÿ;™; q¨"¨bÓ1°1°aÓ8÷	9÷ 	9ò 	9ús   —,AÁAc                 óˆ   — t        j                  d¬«      5  t        j                  |||«      cd d d «       S # 1 sw Y   y xY wrp  )rP   r<  rn   Ú_nct_isfr†	  s       r6   r�   znct_gen._isfÐ  r‹	  ru  c                 óÖ   — t        j                  ||«      }t        j                  ||«      }d|v rt        j                  ||«      nd }d|v rt        j                  ||«      nd }||||fS )Nr  r  )rn   Ú	_nct_meanÚ_nct_varianceÚ_nct_skewnessÚ_nct_kurtosis_excess)rE   r±  rQ  r  rC  rD  rE  rF  s           r6   r   znct_gen._statsÔ  sf   € Ü�]‰]˜2˜rÓ"ˆÜ×Ñ  BÓ'ˆØ*-°©.ŒS×Ñ˜r 2Ô&¸dˆØ14¸±ŒS×%Ñ% b¨"Ô-ÀTˆØ�3˜˜BˆÐr8   r  r  rR	  r‡   r8   r6   r€	  r€	  ”  s5   „ ñò@%òó
-ò
'ò$ò+ò9ò+ôr8   r€	  Únctc                   ót   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	dd„Z
d	„ Ze ee«      ˆ fd
„«       «       Zˆ xZS )Ú
pareto_genaL  A Pareto continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `pareto` is:

    .. math::

        f(x, b) = \frac{b}{x^{b+1}}

    for :math:`x \ge 1`, :math:`b > 0`.

    `pareto` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS rº  rh   rj   s    r6   rk   zpareto_gen._shape_infoõ  r  r8   c                 ó   — ||| dz
  z  z  S r^   r‡   r¼  s      r6   rr   zpareto_gen._pdfø  s   € à�1˜�r˜!‘t‘9‰}Ðr8   c                 ó   — d|| z  z
  S r^   r‡   r¼  s      r6   ru   zpareto_gen._cdfü  s   € Ø�1˜�r‘7‰{Ðr8   c                 ó&   — t        d|z
  d|z  «      S )Nr   r<  r4  rÉ  s      r6   r~   zpareto_gen._ppfÿ  s   € Ü�1�Q‘3˜˜Q™ÓÐr8   c                 ó   — || z  S rN   r‡   r¼  s      r6   ry   zpareto_gen._sf   s   € Ø�A�2‰wˆr8   c                 ó4   — t        j                  |d|z  «      S r?  rÞ  rÉ  s      r6   r�   zpareto_gen._isf   s   € Ü�x‰x˜˜4 !™8Ó$Ð$r8   c                 ó˜  — d\  }}}}d|v rp|dkD  }t        j                  ||«      }t        j                  t        j                  |«      t         j                  ¬«      }t        j
                  ||||dz
  z  «       d|v ry|dkD  }t        j                  ||«      }t        j                  t        j                  |«      t         j                  ¬«      }t        j
                  ||||dz
  z  |dz
  dz  z  «       d	|v r§|d
kD  }t        j                  ||«      }t        j                  t        j                  |«      t         j                  ¬«      }d|dz   z  t        j                  |dz
  «      z  |dz
  t        j                  |«      z  z  }	t        j
                  |||	«       d|v rž|dkD  }t        j                  ||«      }t        j                  t        j                  |«      t         j                  ¬«      }dt        j                  g d¢|«      z  t        j                  g d¢|«      z  }	t        j
                  |||	«       ||||fS )Nrr  r?  r   r•  r‰   rË  rU   r¶   r  r†  rD  r  r$  r�  )r‰   r‰   rÿ  r  )r‰   g      Àrí  rˆ   )	rP   Úextractr—  r·  ri   Úplacer  rÿ   rI  )
rE   rŒ   r  rC  rD  rE  rF  ÚmaskÚbtr  s
             r6   r   zpareto_gen._stats   sÉ  € Ø0‰ˆˆC��RØ�'‰>Ø�q‘5ˆDÜ—‘˜D !Ó$ˆBÜ—‘œŸ™ !›´·±Ô8ˆBÜ�H‰H�R˜˜r R¨¡V™}Ô-Ø�'‰>Ø�q‘5ˆDÜ—‘˜D !Ó$ˆBÜ—'‘'œ"Ÿ(™( 1›+´"·&±&Ô9ˆCÜ�H‰H�S˜$  b¨¡f¡°°C±¸!±Ñ ;Ô<Ø�'‰>Ø�q‘5ˆDÜ—‘˜D !Ó$ˆBÜ—‘œŸ™ !›´·±Ô8ˆBØ˜˜S™‘>¤B§G¡G¨B°©HÓ$5Ñ5¸"¸s¹(ÄbÇgÁgÈbÃkÑ9QÑRˆDÜ�H‰H�R˜˜tÔ$Ø�'‰>Ø�q‘5ˆDÜ—‘˜D !Ó$ˆBÜ—‘œŸ™ !›´·±Ô8ˆBØœŸ
™
Ò#5°rÓ:Ñ:Ü—J‘JÒ5°rÓ:ñ;ˆDä�H‰H�R˜˜tÔ$Ø�3˜˜BˆÐr8   c                 ó>   — dd|z  z   t        j                  |«      z
  S r¿  r2  ©rE   rŒ   s     r6   rò   zpareto_gen._entropy#   ó   € Ø�3�q‘5‰yœ2Ÿ6™6 !›9Ñ$Ð$r8   c                 óø  •‡‡‡‡‡‡‡— t        | ‰||«      }|\  ŠŠ}}|�;t        j                  ‰«      |z
  |xs dk  rt        ddt        j                  ¬«      ‚‰j
                  d   Šˆˆfd„Š||cxu r�€9n �n5ˆfd„Šˆfd„Šˆˆˆˆˆfd„Šˆfd	„}t        |j                  d
d«      «      }|dz  |dz  }
}	 ||	|
«      sE|	dkD  s|
t        j                  k  r-|	dz  }	|
dz  }
 ||	|
«      s|	dkD  rŒ|
t        j                  k  rŒ-t        ‰|	|
g¬«      }|j                  r|j                  }t        j                  ‰«      |z
  }‰xs	  ‰||«      }||z   t        j                  ‰«      k  s.t        j                  ‰«      |z
  }t        j                  |d«      }|||fS t        ‰| �4  ‰fi |¤ŽS |€t        j                  ‰«      |z
  }n|}|xs t        j                  ‰«      |z
  }‰xs	  ‰||«      }|||fS )Nr   Úparetor   rŸ  c                 óf   •— ‰t        j                  t        j                  ‰|z
  | z  «      «      z  S rN   rR  )r/   ÚlocationrF   Úndatas     €€r6   Ú	get_shapez!pareto_gen.fit.<locals>.get_shape3   s+   ø€ ð œ2Ÿ6™6¤"§&¡&¨$°©/¸UÑ)BÓ"CÓDÑDÐDr8   c                 ó   •— ‰| z  |z  S rN   r‡   )r·  r/   r«	  s     €r6   Ú	dL_dScalez!pareto_gen.fit.<locals>.dL_dScale>   s   ø€ ð ˜u‘} uÑ,Ð,r8   c                 óF   •— | dz   t        j                  d‰|z
  z  «      z  S r^   r  )r·  rª	  rF   s     €r6   ÚdL_dLocationz$pareto_gen.fit.<locals>.dL_dLocationC   s&   ø€ ð  ™	¤R§V¡V¨A°¸±Ñ,AÓ%BÑBÐBr8   c                 ót   •— t        j                  ‰«      | z
  }‰xs	  ‰| |«      } ‰||«       ‰|| «      z
  S rN   )rP   r‡  )r/   rª	  r·  r°	  r®	  rF   rO  r¬	  s      €€€€€r6   r   z$pareto_gen.fit.<locals>.fun_to_solveH   sA   ø€ ô Ÿ6™6 $›<¨%Ñ/�ØÒ<¡)¨E°8Ó"<�Ù# E¨8Ó4±yÀÈÓ7NÑNÐNr8   c                 ór   •— t        j                   ‰| «      «      t        j                   ‰|«      «      k7  S rN   rO   ©rR   rS   r   s     €r6   rT   z.pareto_gen.fit.<locals>.interval_contains_rootO   s/   ø€ äŸ™¡¨VÓ 4Ó5ÜŸ™¡¨VÓ 4Ó5ñ6ð 7r8   r/   rU   r!  )rG  rP   r‡  rK  ri   r·  rR  r=   r+   r[  rH  rZ  rA   rC   )rE   rF   rG   r5   r\  rö   r÷   rT   rô  rR   rS   rõ  r/   r.   r·  r°	  r®	  rO  r   r¬	  r«	  r–  s    `             @@@@@@€r6   rC   zpareto_gen.fit&   sø  ÿ€ ô 1°°t¸TÀ4ÓHˆ
Ø%/Ñ"ˆˆf�d˜Fð Ð¤§¡ t£¨tÑ 3°v²{ÀÒ CÜ˜x¨q¼¿¹Ô?Ð?à—
‘
˜1‘ˆõ	Eð
 �6×!Ñ!ô-ô
C÷
Oð Oô7ô   §¡¨°!Ó 4Ó5ˆKØ(¨1™_¨k¸A©o�FˆFñ .¨f°fÔ=Ø š
 f¬r¯v©v¢oØ˜!‘�Ø˜!‘�ñ .¨f°fÔ=Ø ›
 f¬r¯v©v£oô ˜l°V¸VÐ4DÔEˆCØ�}Š}ØŸ™�Ü—f‘f˜T“l UÑ*�ØÒ7¡)¨E°3Ó"7�ð  ™¤r§v¡v¨d£|Ò3ÜŸF™F 4›L¨3Ñ.�EÜŸL™L¨°Ó2�EØ˜c 5Ð(Ð(ä‘w‘{ 4Ñ0¨4Ñ0Ð0Øˆ\Ü—&‘&˜“, Ñ'‰CàˆCð Ò,œ"Ÿ&™& ›,¨Ñ,ˆØÒ/™) E¨3Ó/ˆØ�c˜5Ð Ð r8   r  )rƒ   r„   r…   r†   rk   rr   ru   r~   ry   r�   r   rò   rK   r   r   rC   rÐ  rÑ  s   @r6   r˜	  r˜	  ß  sT   ø„ ñò*Eòòò òò%óò6%ð Ù˜MÓ*óR!ó +ó ôR!r8   r˜	  r¨	  c                   óL   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zy)Ú	lomax_gena§  A Lomax (Pareto of the second kind) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `lomax` is:

    .. math::

        f(x, c) = \frac{c}{(1+x)^{c+1}}

    for :math:`x \ge 0`, :math:`c > 0`.

    `lomax` takes ``c`` as a shape parameter for :math:`c`.

    `lomax` is a special case of `pareto` with ``loc=-1.0``.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zlomax_gen._shape_info˜   r  r8   c                 ó$   — |dz  d|z   |dz   z  z  S r  r‡   r
  s      r6   rr   zlomax_gen._pdf›   s   € à�‰u�c˜!‘e˜q ™uÑ%Ñ%Ð%r8   c                 ód   — t        j                  |«      |dz   t        j                  |«      z  z
  S r^   rþ  r
  s      r6   rÞ   zlomax_gen._logpdfŸ   s&   € Ü�v‰v�a‹y˜A˜a™C¤§¡¨!£Ñ,Ñ,Ð,r8   c                 ó\   — t        j                  | t        j                  |«      z  «       S rN   r  r
  s      r6   ru   zlomax_gen._cdf¢   s"   € Ü—‘˜!˜œBŸH™H Q›K™Ó(Ð(Ð(r8   c                 óZ   — t        j                  | t        j                  |«      z  «      S rN   )rP   r·   rw   r¦  r
  s      r6   ry   zlomax_gen._sf¥   s   € Ü�v‰v�q�bœŸ™ !›‘nÓ%Ð%r8   c                 ó4   — | t        j                  |«      z  S rN   r	  r
  s      r6   rç   zlomax_gen._logsf¨   s   € Øˆr”"—(‘(˜1“+‰~Ðr8   c                 ó\   — t        j                  t        j                  | «       |z  «      S rN   r  r  s      r6   r~   zlomax_gen._ppf«   s!   € Ü�x‰xœŸ™ 1 "›˜ a™Ó(Ð(r8   c                 ó   — |d|z  z  dz
  S r;  r‡   r  s      r6   r�   zlomax_gen._isf®   s   € Ø�4˜!‘8‰}˜qÑ Ð r8   c                 óH   — t         j                  |dd¬«      \  }}}}||||fS )Nr<  rŒ  )r.   r  )r¨	  rò  r�  s         r6   r   zlomax_gen._stats±   s,   € Ü Ÿ,™, q¨d¸F˜,ÓC‰ˆˆC��RØ�3˜˜BˆÐr8   c                 ó>   — dd|z  z   t        j                  |«      z
  S r¿  r2  r~  s     r6   rò   zlomax_gen._entropyµ   s   € Ø��Q‘‰w”r—v‘v˜a“yÑ Ð r8   N)rƒ   r„   r…   r†   rk   rr   rÞ   ru   ry   rç   r~   r�   r   rò   r‡   r8   r6   rµ	  rµ	  €   s:   „ ñò.Eò&ò-ò)ò&òò)ò!òó!r8   rµ	  Úlomaxc                   ó„   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zdd
„Zd„ Ze eed¬«      ˆ fd„«       «       Zˆ xZS )Úpearson3_gena�  A pearson type III continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `pearson3` is:

    .. math::

        f(x, \kappa) = \frac{|\beta|}{\Gamma(\alpha)}
                       (\beta (x - \zeta))^{\alpha - 1}
                       \exp(-\beta (x - \zeta))

    where:

    .. math::

            \beta = \frac{2}{\kappa}

            \alpha = \beta^2 = \frac{4}{\kappa^2}

            \zeta = -\frac{\alpha}{\beta} = -\beta

    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).
    Pass the skew :math:`\kappa` into `pearson3` as the shape parameter
    ``skew``.

    %(after_notes)s

    %(example)s

    References
    ----------
    R.W. Vogel and D.E. McMartin, "Probability Plot Goodness-of-Fit and
    Skewness Estimation Procedures for the Pearson Type 3 Distribution", Water
    Resources Research, Vol.27, 3149-3158 (1991).

    L.R. Salvosa, "Tables of Pearson's Type III Function", Ann. Math. Statist.,
    Vol.1, 191-198 (1930).

    "Using Modern Computing Tools to Fit the Pearson Type III Distribution to
    Aviation Loads Data", Office of Aviation Research (2003).

    c                 óú   — d}d}d}t        j                  d||«      \  }}}|j                  «       }t        j                  |«      |k  }| }d||   |z  z  }	||	z  dz  }
||
|	z  z
  }|	||   |z
  z  }||||||	|
|fS )Nrˆ   r‰   g�íµ ÷Æð>r¶   rU   )rP   rñ  Úcopyr  )rE   rq   rA  r.   r/   Únorm2pearson_transitionÚansr¢	  Úinvmaskrn  r  r�  Útransxs                r6   Ú_preprocesszpearson3_gen._preprocessê   s±   € ð
 ˆØˆð #+Ðä×*Ñ*¨3°°4Ó8‰ˆˆQ�Ø�h‰h‹jˆô �{‰{˜4Ó Ð#:Ñ:ˆØ�%ˆà�d˜7‘m eÑ+Ñ,ˆØ˜‘ Ñ!ˆØ�U˜T‘\Ñ!ˆà˜˜7™ dÑ*Ñ+ˆØ�A�v˜t W¨d°E¸4Ð?Ð?r8   c                 ó,   — t        j                  |«      S rN   r™  )rE   rA  s     r6   rc   zpearson3_gen._argcheck!  s   € ô
 �{‰{˜4Ó Ð r8   c                 ó^   — t        ddt        j                   t        j                  fd«      gS )NrA  Fr
  rh   rj   s    r6   rk   zpearson3_gen._shape_info!  s%   € Ü˜6 5¬B¯F©F¨7´B·F±FÐ*;¸^ÓLÐMÐMr8   c                 ó*   — d}d}|}d|dz  z  }||||fS )Nrˆ   r‰   rÊ  rU   r‡   )rE   rA  r?  rË  r  r  s         r6   r   zpearson3_gen._stats!  s,   € ØˆØˆØˆØ��a‘‰KˆØ�!�Q˜ˆzÐr8   c                 óÎ   — t        j                  | j                  ||«      «      }|j                  dk(  rt        j                  |«      ry|S d|t        j                  |«      <   |S )Nr   rˆ   )rP   r·   rÞ   r+  r¢  )rE   rq   rA  rÆ	  s       r6   rr   zpearson3_gen._pdf!  sR   € ô
 �f‰f�T—\‘\ ! TÓ*Ó+ˆØ�8‰8�qŠ=Ü�x‰x˜Œ}ØØˆJØ ˆŒB�H‰H�S‹MÑØˆ
r8   c                 óô   — | j                  ||«      \  }}}}}}}}	t        j                  t        ||   «      «      ||<   t        j                  t	        |«      «      t
        j                  ||«      z   ||<   |S rN   )rÉ	  rP   rð   rº   rŽ  rØ  ré  )
rE   rq   rA  rÆ	  rÈ	  r¢	  rÇ	  rn  r  r	  s
             r6   rÞ   zpearson3_gen._logpdf$!  ss   € ð ×Ñ˜Q Ó%ñ 	6ˆˆQ�˜˜g t¨U°Aô —F‘Fœ9 Q t¡WÓ-Ó.ˆˆD‰	ô —v‘vœc $›iÓ(¬5¯<©<¸ÀÓ+FÑFˆˆG‰Øˆ
r8   c                 óž  — | j                  ||«      \  }}}}}}}}t        ||   «      ||<   t        j                  ||j                  «      }t        j
                  ||dkD  «      }	||   dkD  }
t        j                  ||
   ||
   «      ||	<   t        j
                  ||dk  «      }||   dk  }t        j                  ||   ||   «      ||<   |S r2  )	rÉ	  rÀ   rP   rm	  r·  rƒ  rØ  r�   rÕ  ©rE   rq   rA  rÆ	  rÈ	  r¢	  rÇ	  r	  r  Ú	invmask1aÚ	invmask1bÚ	invmask2aÚ	invmask2bs                r6   ru   zpearson3_gen._cdf3!  sØ   € à×Ñ˜Q Ó%ñ 	3ˆˆQ�˜˜g q¨%°ô ˜a ™gÓ&ˆˆD‰	ä�‰˜t W§]¡]Ó3ˆÜ—N‘N 7¨D°1©HÓ5ˆ	Ø˜‘M AÑ%ˆ	ô Ÿ™ 6¨)Ñ#4°e¸IÑ6FÓGˆˆI‰ô —N‘N 7¨D°1©HÓ5ˆ	Ø˜‘M AÑ%ˆ	äŸ™ &¨Ñ"3°U¸9Ñ5EÓFˆˆI‰àˆ
r8   c                 óž  — | j                  ||«      \  }}}}}}}}t        ||   «      ||<   t        j                  ||j                  «      }t        j
                  ||dkD  «      }	||   dkD  }
t        j                  ||
   ||
   «      ||	<   t        j
                  ||dk  «      }||   dk  }t        j                  ||   ||   «      ||<   |S r2  )	rÉ	  rÊ   rP   rm	  r·  rƒ  rØ  rÕ  r�   rÐ	  s                r6   ry   zpearson3_gen._sfK!  sÔ   € à×Ñ˜Q Ó%ñ 	3ˆˆQ�˜˜g q¨%°ô ˜Q˜t™WÓ%ˆˆD‰	ä�‰˜t W§]¡]Ó3ˆÜ—N‘N 7¨D°1©HÓ5ˆ	Ø˜‘M AÑ%ˆ	ÜŸ™ &¨Ñ"3°U¸9Ñ5EÓFˆˆI‰ä—N‘N 7¨D°1©HÓ5ˆ	Ø˜‘M AÑ%ˆ	ÜŸ™ 6¨)Ñ#4°e¸IÑ6FÓGˆˆI‰àˆ
r8   c                 ó  — t        j                  ||«      }| j                  dg|«      \  }}}}}}}	}
|j                  «       }|j                  |z
  }|j                  |«      ||<   |j                  |	|«      |z  |
z   ||<   |dk(  r|d   }|S )Nr   r‡   )rP   rm	  rÉ	  r¥  r×   rÕ   r%  )rE   rA  r×   rØ   rÆ	  r	  r¢	  rÇ	  rn  r  r�  ÚnsmallÚnbigs                r6   rÙ   zpearson3_gen._rvs\!  s�   € Ü�‰˜t TÓ*ˆà×Ñ˜a˜S $Ó'ñ 	4ˆˆQ��4˜ $¨¨tð —‘“ˆØ�y‰y˜6Ñ!ˆØ ×0Ñ0°Ó8ˆˆD‰	Ø#×2Ñ2°5¸$Ó?ÀÑDÀtÑKˆˆG‰à�2Š:Ø�a‘&ˆCØˆ
r8   c                 óÈ   — | j                  ||«      \  }}}}}}}}	t        ||   «      ||<   ||   }d||dk     z
  ||dk  <   t        j                  ||«      |z  |	z   ||<   |S r�  )rÉ	  rÇ   rw   rÄ  )
rE   r}   rA  rÆ	  r	  r¢	  rÇ	  rn  r  r�  s
             r6   r~   zpearson3_gen._ppfj!  s~   € à×Ñ˜Q Ó%ñ 	4ˆˆQ��4˜ $¨¨tä˜a ™gÓ&ˆˆD‰	Øˆg‰JˆØ˜!˜D 1™H™+‘oˆˆ$�‰(‰Ü—~‘~ e¨QÓ/°Ñ4°tÑ;ˆˆG‰Øˆ
r8   ze        Note that method of moments (`method='MM'`) is not
        available for this distribution.

ró   c                 ó|   •— |j                  dd «      dk(  rt        d«      ‚t        t        | «      | �  |g|¢­i |¤ŽS )Nr1   ÚMMzhFit `method='MM'` is not available for the Pearson3 distribution. Please try the default `method='MLE'`.)r=   ÚNotImplementedErrorrA   rB   rC   rP  s       €r6   rC   zpearson3_gen.fits!  sO   ø€ ð
 �8‰8�H˜dÓ# tÒ+Ü%ð 'Dó Eð Eô œ˜d› TÑ.¨tÐC°dÒC¸dÑCÐCr8   r  )rƒ   r„   r…   r†   rÉ	  rc   rk   r   rr   rÞ   ru   ry   rÙ   r~   rK   r	   r   rC   rÐ  rÑ  s   @r6   rÂ	  rÂ	  ¼   sg   ø„ ñ,òZ@ò8!òNòòòòò0ó"òð Ù˜}ð 50ô 1óDó1ó ôDr8   rÂ	  Úpearson3c                   óŒ   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zˆ fd„Ze eed¬«      ˆ fd„«       «       Zˆ xZS )Úpowerlaw_gena¨  A power-function continuous random variable.

    %(before_notes)s

    See Also
    --------
    pareto

    Notes
    -----
    The probability density function for `powerlaw` is:

    .. math::

        f(x, a) = a x^{a-1}

    for :math:`0 \le x \le 1`, :math:`a > 0`.

    `powerlaw` takes ``a`` as a shape parameter for :math:`a`.

    %(after_notes)s

    For example, the support of `powerlaw` can be adjusted from the default
    interval ``[0, 1]`` to the interval ``[c, c+d]`` by setting ``loc=c`` and
    ``scale=d``. For a power-law distribution with infinite support, see
    `pareto`.

    `powerlaw` is a special case of `beta` with ``b=1``.

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r	  rh   rj   s    r6   rk   zpowerlaw_gen._shape_info¤!  r  r8   c                 ó   — |||dz
  z  z  S r  r‡   r  s      r6   rr   zpowerlaw_gen._pdf§!  s   € à��Q�s‘U‘‰|Ðr8   c                 ó`   — t        j                  |«      t        j                  |dz
  |«      z   S r^   )rP   rð   rw   rx  r  s      r6   rÞ   zpowerlaw_gen._logpdf«!  s$   € Ü�v‰v�a‹yœ2Ÿ8™8 A¨¡E¨1Ó-Ñ-Ð-r8   c                 ó   — ||dz  z  S r  r‡   r  s      r6   ru   zpowerlaw_gen._cdf®!  s   € Ø�1�S‘5‰zÐr8   c                 ó2   — |t        j                  |«      z  S rN   r2  r  s      r6   rã   zpowerlaw_gen._logcdf±!  r3  r8   c                 ó    — t        |d|z  «      S r  r4  r  s      r6   r~   zpowerlaw_gen._ppf´!  s   € Ü�1�c˜!‘e‹}Ðr8   c                 ó0   — t        j                  ||«       S rN   )rw   r‡  )rE   rô  r‹   s      r6   ry   zpowerlaw_gen._sf·!  s   € Ü—‘˜˜A“ˆÐr8   c                 ó   — |||z   z  S rN   r‡   r0  s      r6   r  zpowerlaw_gen._munpº!  s   € à�A˜‘E‰{Ðr8   c                 óØ   — ||dz   z  ||dz   z  |dz   dz  z  d|dz
  |dz   z  z  t        j                  |dz   |z  «      z  dt        j                  g d¢|«      z  ||dz   z  |dz   z  z  fS )	Nr‰   r¶   rU   r³  rD  r…  )r   r¿  rÿ  rU   r$  )rP   rÿ   rI  r  s     r6   r   zpowerlaw_gen._stats¾!  s…   € Ø�Q˜‘W‘Ø�Q˜‘W‘  S¡¨Q¡Ñ.Ø˜˜S™ Q¨¡WÑ-Ñ.´·±¸!¸c¹'ÀQ¹Ó1GÑGØ”B—J‘Jš~¨qÓ1Ñ1°Q¸!¸c¹'±]ÀaÈ!ÁeÑ5LÑMðOð 	Or8   c                 ó>   — dd|z  z
  t        j                  |«      z
  S r¿  r2  r  s     r6   rò   zpowerlaw_gen._entropyÄ!  r¦	  r8   c                 ó<   •— t         ‰| �  ||«      |dk7  |dk\  z  z  S r”  )rA   r  )rE   rq   r‹   r–  s      €r6   r  zpowerlaw_gen._support_maskÇ!  s,   ø€ Ü‘Ñ% a¨Ó+Ø˜‘F˜q A™vÑ&ñ(ð 	)r8   a:          Notes specifically for ``powerlaw.fit``: If the location is a free
        parameter and the value returned for the shape parameter is less than
        one, the true maximum likelihood approaches infinity. This causes
        numerical difficulties, and the resulting estimates are approximate.
        

ró   c                 óR  •‡‡‡‡‡‡‡‡— |j                  dd«      rt        ‰| �  ‰g|¢­i |¤ŽS t        t	        j
                  ‰«      «      dk(  rt        ‰| �  ‰g|¢­i |¤ŽS t        | ‰||«      \  ŠŠ}}‰| j                  ‰«      fg}| j                  |i «      d   }|�E‰j                  «       |kD  st        ddd«      ‚|�#‰j                  «       ||z   k  st        ddd«      ‚|�5|dk  rt        d«      ‚|t	        j                  ‰«      k  rd}t        |«      ‚d„ Šd	„ Š|�|� ‰‰||«      ||fS |�«t	        j                  ‰j                  «       t        j                   «      }	‰xs
  ‰‰|	|«      }
 ||
|	|f‰«      }t	        j                  ‰j                  «       |z
  t        j                  «      }‰xs
  ‰‰||«      } ||||f‰«      }||k  r|
|	|fS |||fS |� ‰‰|«      }‰xs
  ‰‰||«      }|||fS ˆˆˆˆfd
„}d„ Šd„ Šˆˆˆˆˆfd„Šˆˆˆˆˆˆfd„Šˆˆˆˆˆˆfd„}‰�‰dk  r |«       S ‰�‰dkD  r |«       S  |«       }| j!                  |‰«      } |«       }| j!                  |‰«      }||k  r
|d   dk  r|S ||kD  r
|d   dkD  r|S t        ‰| �  ‰g|¢­i |¤ŽS )Nr;  Fr   Úpowerlawr   zKNegative or zero `fscale` is outside the range allowed by the distribution.z0`fscale` must be greater than the range of data.c                 ó¨   — t        | «      }| t        j                  t        j                  | |z
  «      «      |t        j                  |«      z  z
  z  S rN   )r¤  rP   r¥  rð   )rF   r.   r/   rë  s       r6   r¬	  z#powerlaw_gen.fit.<locals>.get_shape"  sA   € ô �D“	ˆAØ�3œ"Ÿ&™&¤§¡¨¨s©
Ó!3Ó4°q¼¿¹À»±ÑFÑGÐGr8   c                 ó(   — | j                  «       |z
  S rN   )r£  )rF   r.   s     r6   Ú	get_scalez#powerlaw_gen.fit.<locals>.get_scale"  s   € ð —8‘8“: Ñ#Ð#r8   c                  óâ  •— t        j                  ‰j                  «       t         j                   «      } t        j                  | «      t        j
                  | j                  «      j                  k  r?t        j                  | «      t        j
                  | j                  «      j                  z  } t        j                   ‰‰| «      t         j                  «      }‰xs
  ‰‰| |«      }|| |fS rN   )	rP   rZ  r‡  ri   rŽ  r#  r	  r$  rQ   )r.   r/   r·  rF   rO  rï	  r¬	  s      €€€€r6   Úfit_loc_scale_w_shape_lt_1z4powerlaw_gen.fit.<locals>.fit_loc_scale_w_shape_lt_1="  s¢   ø€ Ü—,‘,˜tŸx™x›z¬B¯F©F¨7Ó3ˆCÜ�v‰v�c‹{œRŸX™X c§i¡iÓ0×5Ñ5Ò5Ü—g‘g˜c“l¤R§X¡X¨c¯i©iÓ%8×%=Ñ%=Ñ=�Ü—L‘L¡¨4°Ó!5´r·v±vÓ>ˆEØÒ9™i¨¨c°5Ó9ˆEØ˜#˜uÐ$Ð$r8   c                 ó.   — | j                   d    |z  |z  S r2  )r·  )rF   r·  r/   s      r6   r®	  z#powerlaw_gen.fit.<locals>.dL_dScaleL"  s   € ð —J‘J˜q‘M�> EÑ)¨EÑ1Ð1r8   c                 óD   — |dz
  t        j                  d|| z
  z  «      z  S r^   r  )rF   r·  r.   s      r6   r°	  z&powerlaw_gen.fit.<locals>.dL_dLocationQ"  s%   € ð ˜A‘I¤§¡¨¨S°4©ZÑ(8Ó!9Ñ9Ð9r8   c                 óŒ   •— t        j                   ‰‰| «      t         j                   «      }‰xs
  ‰‰| |«      } ‰‰|| «      S rN   ©rP   rZ  ri   )r.   r/   r·  r°	  rF   rO  rï	  r¬	  s      €€€€€r6   ÚdL_dLocation_starz+powerlaw_gen.fit.<locals>.dL_dLocation_starV"  sD   ø€ ô —L‘L¡¨4°Ó!5¼¿¹°wÓ?ˆEØÒ9™i¨¨c°5Ó9ˆEÙ  e¨SÓ1Ð1r8   c                 ó¢   •— t        j                   ‰‰| «      t         j                   «      }‰xs
  ‰‰| |«      } ‰‰||«       ‰‰|| «      z
  S rN   rõ	  )	r.   r/   r·  r°	  r®	  rF   rO  rï	  r¬	  s	      €€€€€€r6   r   z&powerlaw_gen.fit.<locals>.fun_to_solve]"  sW   ø€ ô —L‘L¡¨4°Ó!5¼¿¹°wÓ?ˆEØÒ9™i¨¨c°5Ó9ˆEÙ˜d E¨5Ó1Ù" 4¨°Ó4ñ5ð 6r8   c                  ó¸  •— t        j                  ‰
j                  «       t         j                   «      } ‰
j                  «       | z
  } ‰	| «      dkD  r$‰
j                  «       |z
  } |dz  } ‰	| «      dkD  rŒ$ˆfd„}| dz
  }d} ||| «      sJ|t         j                   k7  r6‰
j                  «       |z
  }|dz  } ||| «      s|t         j                   k7  rŒ6t	        j
                  ‰|| f¬«      }t        j                  |j                  t         j                   «      }t        j                   ‰‰
|«      t         j                  «      }‰xs
  ‰‰
||«      }|||fS )Nr   rU   c                 ór   •— t        j                   ‰| «      «      t        j                   ‰|«      «      k7  S rN   rO   r³	  s     €r6   rT   zTpowerlaw_gen.fit.<locals>.fit_loc_scale_w_shape_gt_1.<locals>.interval_contains_rootq"  s/   ø€ äŸ™¡¨VÓ 4Ó5ÜŸ7™7¡<°Ó#7Ó8ñ9ð :r8   r   r‰   r!  )rP   rZ  r‡  ri   r   r+   rH  )rS   r%  rT   rR   rþ  rH  r.   r/   r·  rö	  rF   rO  r   rï	  r¬	  s            €€€€€€r6   Úfit_loc_scale_w_shape_gt_1z4powerlaw_gen.fit.<locals>.fit_loc_scale_w_shape_gt_1e"  s7  ø€ ô —\‘\ $§(¡(£*¬r¯v©v¨gÓ6ˆFð —X‘X“Z &Ñ(ˆEÙ# FÓ+¨aÒ/ØŸ™› eÑ+�Ø˜‘
�ñ $ FÓ+¨aÓ/ô:ð
 ˜a‘ZˆFð
 ˆAÙ-¨f°fÔ=Ø¤"§&¡& Ò(ØŸ(™(›* q™.�Ø�Q‘�ñ .¨f°fÔ=Ø¤"§&¡& Ó(ô ×'Ñ'¨¸vÀvÐ>NÔOˆDä—,‘,˜tŸy™y¬2¯6©6¨'Ó2ˆCÜ—L‘L¡¨4°Ó!5´r·v±vÓ>ˆEØÒ9™i¨¨c°5Ó9ˆEØ˜#˜uÐ$Ð$r8   )r3   rA   rC   r¤  rP   ÚuniquerG  r•  Ú_reduce_funcr‡  rK  r£  rú   ÚptprZ  ri   rV  )rE   rF   rG   r5   rö   r÷   Úpenalized_nllf_argsÚpenalized_nllfrY   Úloc_lt1Ú	shape_lt1Úll_lt1Úloc_gt1Ú	shape_gt1Úll_gt1r/   r·  rñ	  rú	  Úfit_shape_lt1Úfit_shape_gt1r°	  rö	  r®	  rO  r   rï	  r¬	  r–  s    `                   @@@@@@@€r6   rC   zpowerlaw_gen.fitË!  s   ÿø€ ðP �8‰8�J Ô&Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3äŒr�y‰y˜‹Ó 1Ò$Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ä%@ÀÀtØAEÀtó&MÑ"ˆˆf�d˜Fà# d§n¡n°TÓ&:Ð%<Ð=ÐØ×*Ñ*Ð+>ÀÓCÀAÑFˆð
 ÐØ—8‘8“: Ò$Ü" :¨q°!Ó4Ð4ØÐ!¨$¯(©(«*¸¸v¹Ò*EÜ" :¨q°!Ó4Ð4àÐØ˜Š{Ü ð "Fó Gð GàœŸ™ ›Ò%ØH�Ü  “oÐ%ò	Hò	$ð Ð $Ð"2Ù˜T 4¨Ó0°$¸Ð>Ð>ð Ðä—l‘l 4§8¡8£:´·±¨wÓ7ˆGØÒB¡)¨D°'¸6Ó"BˆIÙ# Y°¸Ð$@À$ÓGˆFô —l‘l 4§8¡8£:°Ñ#6¼¿¹Ó?ˆGØÒB¡)¨D°'¸6Ó"BˆIÙ# Y°¸Ð$@À$ÓGˆFà˜ŠØ  '¨6Ð1Ð1à  '¨6Ð1Ð1ð ÐÙ˜d DÓ)ˆEØÒ:™i¨¨d°EÓ:ˆEØ˜$ Ð%Ð%÷
	%ò	2ò
	:÷
	2ð 	2÷	6ñ 	6÷!	%ñ !	%ðH Ð &¨A¢+Ù-Ó/Ð/ØÐ F¨Q¢JÙ-Ó/Ð/ñ 3Ó4ˆØ—‘˜=¨$Ó/ˆá2Ó4ˆØ—‘˜=¨$Ó/ˆà�VÒ ¨aÑ 0°AÒ 5Ø Ð Ø�fŠ_ ¨qÑ!1°AÒ!5Ø Ð ä‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3r8   )rƒ   r„   r…   r†   rk   rr   rÞ   ru   rã   r~   ry   r  r   rò   r  rK   r	   r   rC   rÐ  rÑ  s   @r6   rß	  rß	  ƒ!  sl   ø„ ñò@Eòò.òòòòòòOò%ô)ð Ù˜}ð 5ô óH4óó ôH4r8   rß	  rì	  c                   óX   — e Zd ZdZej
                  Zd„ Zd„ Zd„ Z	d„ Z
d„ Zd„ Zd„ Zd	„ Zy
)Úpowerlognorm_genañ  A power log-normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `powerlognorm` is:

    .. math::

        f(x, c, s) = \frac{c}{x s} \phi(\log(x)/s)
                     (\Phi(-\log(x)/s))^{c-1}

    where :math:`\phi` is the normal pdf, and :math:`\Phi` is the normal cdf,
    and :math:`x > 0`, :math:`s, c > 0`.

    `powerlognorm` takes :math:`c` and :math:`s` as shape parameters.

    %(after_notes)s

    %(example)s

    c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS )Nr  Fr   r
  r  rh   )rE   r!  r‡  s      r6   rk   zpowerlognorm_gen._shape_infoº"  rˆ  r8   c                 óN   — t        j                  | j                  |||«      «      S rN   rÝ  ©rE   rq   r  r  s       r6   rr   zpowerlognorm_gen._pdf¿"  rY  r8   c                 ó  — t        j                  |«      t        j                  |«      z
  t        j                  |«      z
  t        t        j                  |«      |z  «      z   t        t        j                  |«       |z  «      |dz
  z  z   S r  ©rP   rð   r½   rÃ   r
  s       r6   rÞ   zpowerlognorm_gen._logpdfÂ"  si   € Ü—‘�q“	œBŸF™F 1›IÑ%¬¯©¨q«	Ñ1ÜœRŸV™V A›Y¨™]Ó+ñ,äœbŸf™f Q›i˜Z¨!™^Ó,°°B±Ñ7ñ8ð 	9r8   c                 óP   — t        j                  | j                  |||«      «       S rN   ra  r
  s       r6   ru   zpowerlognorm_gen._cdfÇ"  rb  r8   c                 ó.   — | j                  d|z
  ||«      S r^   )r�   ©rE   r}   r  r  s       r6   r~   zpowerlognorm_gen._ppfÊ"  s   € Ø�y‰y˜˜Q™  1Ó%Ð%r8   c                 óN   — t        j                  | j                  |||«      «      S rN   r6  r
  s       r6   ry   zpowerlognorm_gen._sfÍ"  r7  r8   c                 óL   — t        t        j                  |«       |z  «      |z  S rN   r”  r
  s       r6   rç   zpowerlognorm_gen._logsfÐ"  s    € ÜœRŸV™V A›Y˜J¨™NÓ+¨aÑ/Ð/r8   c                 óR   — t        j                  t        |d|z  z  «       |z  «      S r^   r?  r
  s       r6   r�   zpowerlognorm_gen._isfÓ"  s&   € Ü�v‰v”y  Q q¡S¡Ó*Ð*¨QÑ.Ó/Ð/r8   N)rƒ   r„   r…   r†   r   r  r  rk   rr   rÞ   ru   r~   ry   rç   r�   r‡   r8   r6   r	
  r	
   "  s<   „ ñð. "×4Ñ4€Mòò
-ò9ò
/ò&ò,ò0ó0r8   r	
  Úpowerlognormc                   ó@   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zy
)Úpowernorm_genah  A power normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `powernorm` is:

    .. math::

        f(x, c) = c \phi(x) (\Phi(-x))^{c-1}

    where :math:`\phi` is the normal pdf, :math:`\Phi` is the normal cdf,
    :math:`x` is any real, and :math:`c > 0` [1]_.

    `powernorm` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    References
    ----------
    .. [1] NIST Engineering Statistics Handbook, Section 1.3.6.6.13,
           https://www.itl.nist.gov/div898/handbook//eda/section3/eda366d.htm

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zpowernorm_gen._shape_infoö"  r  r8   c                 óD   — |t        |«      z  t        | «      |dz
  z  z  S r  ©rº   rÀ   r
  s      r6   rr   zpowernorm_gen._pdfù"  s$   € à”˜1“‰~¤¨A¨2£°°3±Ñ!7Ñ8Ð8r8   c                 ój   — t        j                  |«      t        |«      z   |dz
  t        | «      z  z   S r^   r
  r
  s      r6   rÞ   zpowernorm_gen._logpdfý"  s.   € Ü�v‰v�a‹yœ<¨›?Ñ*¨a°©c´<ÀÀÓ3CÑ-CÑCÐCr8   c                 óN   — t        j                  | j                  ||«      «       S rN   ra  r
  s      r6   ru   zpowernorm_gen._cdf #  s   € Ü—‘˜Ÿ™ Q¨Ó*Ó+Ð+Ð+r8   c                 ó:   — t        t        d|z
  d|z  «      «       S r  )rÇ   rÈ  r  s      r6   r~   zpowernorm_gen._ppf#  s   € Üœ#˜c A™g s¨Q¡wÓ/Ó0Ð0Ð0r8   c                 óL   — t        j                  | j                  ||«      «      S rN   r6  r
  s      r6   ry   zpowernorm_gen._sf#  r2  r8   c                 ó    — |t        | «      z  S rN   rÌ   r
  s      r6   rç   zpowernorm_gen._logsf	#  s   € Ø”<  Ó#Ñ#Ð#r8   c                 ól   — t        t        j                  t        j                  |«      |z  «      «       S rN   )rÇ   rP   r·   rð   r  s      r6   r�   zpowernorm_gen._isf#  s%   € Üœ"Ÿ&™&¤§¡¨£¨Q¡Ó/Ó0Ð0Ð0r8   N)rƒ   r„   r…   r†   rk   rr   rÞ   ru   r~   ry   rç   r�   r‡   r8   r6   r
  r
  Ú"  s1   „ ñò6Eò9òDò,ò1ò)ò$ó1r8   r
  Ú	powernormc                   óB   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	dd	„Z
d
„ Zy)Ú	rdist_gena/  An R-distributed (symmetric beta) continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `rdist` is:

    .. math::

        f(x, c) = \frac{(1-x^2)^{c/2-1}}{B(1/2, c/2)}

    for :math:`-1 \le x \le 1`, :math:`c > 0`. `rdist` is also called the
    symmetric beta distribution: if B has a `beta` distribution with
    parameters (c/2, c/2), then X = 2*B - 1 follows a R-distribution with
    parameter c.

    `rdist` takes ``c`` as a shape parameter for :math:`c`.

    This distribution includes the following distribution kernels as
    special cases::

        c = 2:  uniform
        c = 3:  `semicircular`
        c = 4:  Epanechnikov (parabolic)
        c = 6:  quartic (biweight)
        c = 8:  triweight

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r  rh   rj   s    r6   rk   zrdist_gen._shape_info5#  r  r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  r
  s      r6   rr   zrdist_gen._pdf9#  r‘  r8   c                 óv   — t        j                  d«       t        j                  |dz   dz  |dz  |dz  «      z   S r‹  )rP   rð   rn  rÞ   r
  s      r6   rÞ   zrdist_gen._logpdf<#  s4   € Ü—‘�q“	ˆzœDŸL™L¨!¨a©%°©°A°a±C¸¸1¹Ó=Ñ=Ð=r8   c                 óH   — t         j                  |dz   dz  |dz  |dz  «      S r1  ré  r
  s      r6   ru   zrdist_gen._cdf?#  s%   € Ü�y‰y˜!˜a™% ™ A a¡C¨¨1©Ó-Ð-r8   c                 óH   — t         j                  |dz   dz  |dz  |dz  «      S r1  râ  r
  s      r6   ry   zrdist_gen._sfB#  s%   € Ü�x‰x˜˜Q™ ™	 1 Q¡3¨¨!©Ó,Ð,r8   c                 óH   — dt         j                  ||dz  |dz  «      z  dz
  S r‹  )rn  r~   r  s      r6   r~   zrdist_gen._ppfE#  s'   € Ø”—‘˜1˜a ™c 1 Q¡3Ó'Ñ'¨!Ñ+Ð+r8   Nc                 ó@   — d|j                  |dz  |dz  |«      z  dz
  S r‹  rm  rë  s       r6   rÙ   zrdist_gen._rvsH#  s)   € Ø�<×$Ñ$ Q q¡S¨!¨A©#¨tÓ4Ñ4°qÑ8Ð8r8   c                 óŠ   — d|dz  z
  t        j                  |dz   dz  |dz  «      z  }|t        j                  d|dz  «      z  S )Nr   rU   r‰   r¶   r”   rT  )rE   rb   r  Ú	numerators       r6   r  zrdist_gen._munpK#  sE   € Ø˜!˜a™%‘[¤B§G¡G¨Q°©W¸©M¸1¸s¹7Ó$CÑCˆ	Øœ2Ÿ7™7 6¨1¨r©6Ó2Ñ2Ð2r8   r  )rƒ   r„   r…   r†   rk   rr   rÞ   ru   ry   r~   rÙ   r  r‡   r8   r6   r#
  r#
  #  s1   „ ñ òBEò*ò>ò.ò-ò,ó9ó3r8   r#
  r<  Úrdistc                   ó¢   ‡ — e Zd ZdZej
                  Zd„ Zdd„Zd„ Z	d„ Z
d„ Zd„ Zd„ Zd	„ Zd
„ Zd„ Zd„ Ze eed¬«      ˆ fd„«       «       Zˆ xZS )Úrayleigh_gena7  A Rayleigh continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `rayleigh` is:

    .. math::

        f(x) = x \exp(-x^2/2)

    for :math:`x \ge 0`.

    `rayleigh` is a special case of `chi` with ``df=2``.

    %(after_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zrayleigh_gen._shape_infok#  r¦   r8   c                 ó2   — t         j                  d||¬«      S )NrU   r×  rt  rÖ   s      r6   rÙ   zrayleigh_gen._rvsn#  s   € Ü�w‰w�q˜t°,ˆwÓ?Ð?r8   c                 óJ   — t        j                  | j                  |«      «      S rN   rÝ  ©rE   rõ  s     r6   rr   zrayleigh_gen._pdfq#  rÞ  r8   c                 ó>   — t        j                  |«      d|z  |z  z
  S r  r2  r3
  s     r6   rÞ   zrayleigh_gen._logpdfu#  s   € Ü�v‰v�a‹y˜3 ™7 Q™;Ñ&Ð&r8   c                 ó:   — t        j                  d|dz  z  «       S rä  ry  r3
  s     r6   ru   zrayleigh_gen._cdfx#  s   € Ü—‘˜  1¡™Ó%Ð%Ð%r8   c                 óZ   — t        j                  dt        j                  | «      z  «      S ©Nr  )rP   rÿ   rw   r¦  r°   s     r6   r~   zrayleigh_gen._ppf{#  s    € Ü�w‰w�rœBŸH™H a R›LÑ(Ó)Ð)r8   c                 óJ   — t        j                  | j                  |«      «      S rN   r6  r3
  s     r6   ry   zrayleigh_gen._sf~#  s   € Ü�v‰v�d—k‘k !“nÓ%Ð%r8   c                 ó   — d|z  |z  S )Nr$  r‡   r3
  s     r6   rç   zrayleigh_gen._logsf�#  s   € Ø�a‰x˜!‰|Ðr8   c                 óX   — t        j                  dt        j                  |«      z  «      S r7
  )rP   rÿ   rð   r°   s     r6   r�   zrayleigh_gen._isf„#  s   € Ü�w‰w�rœBŸF™F 1›I‘~Ó&Ð&r8   c                 ó8  — dt         j                  z
  }t        j                  t         j                  dz  «      |dz  dt         j                  dz
  z  t        j                  t         j                  «      z  |dz  z  dt         j                  z  |z  d|dz  z  z
  fS )Nr$  rU   r†  rÊ  r…  r,  r  r€  s     r6   r   zrayleigh_gen._stats‡#  sy   € Ø”"—%‘%‰iˆÜ—‘œŸ™˜a™Ó Ø�A‘Ø”2—5‘5˜‘7‘œBŸG™G¤B§E¡E›NÑ*¨3°©8Ñ3Ø”"—%‘%‘˜‘˜B˜s A™v™IÑ%ð'ð 	'r8   c                 óL   — t         dz  dz   dt        j                  d«      z  z
  S )Nr¶   r   r”   rU   r8  rj   s    r6   rò   zrayleigh_gen._entropyŽ#  s!   € Ü�c‰z˜A‰~ ¤B§F¡F¨1£I¡Ñ-Ð-r8   aú          Notes specifically for ``rayleigh.fit``: If the location is fixed with
        the `floc` parameter, this method uses an analytical formula to find
        the scale.  Otherwise, this function uses a numerical root finder on
        the first order conditions of the log-likelihood function to find the
        MLE.  Only the (optional) `loc` parameter is used as the initial guess
        for the root finder; the `scale` parameter and any other parameters
        for the optimizer are ignored.

ró   c                 óª  •‡— |j                  dd«      rt        ‰| �  ‰g|¢­i |¤ŽS t        | ‰||«      \  Š}}ˆfd„}ˆfd„}|fˆfd„	}|�At	        j
                  ‰|z
  dk  «      rt        ddt        j                  ¬	«      ‚| ||«      fS |j                  d
«      }	|	€| j                  ‰«      d   }	|€|n|}
t	        j                  t	        j                  ‰«      t        j                   «      }t        |
|«      }t        j                  |
||f¬«      }|j                  st!        |j"                  «      ‚|j$                  }|xs  ||«      }||fS )Nr;  Fc                 ó^   •— t        j                  ‰| z
  dz  «      dt        ‰«      z  z  dz  S rÂ  )rP   r¥  r¤  )r.   rF   s    €r6   Ú	scale_mlez#rayleigh_gen.fit.<locals>.scale_mle #  s/   ø€ ô —F‘F˜D 3™J¨1Ñ,Ó-°´S¸³Y±Ñ?ÀBÑFÐFr8   c                 ó¨   •— ‰| z
  }|j                  «       }|dz  j                  «       }d|z  j                  «       }||dt        ‰«      z  z  |z  z
  S r‹  )r¥  r¤  )r.   r#  r\  rb  Ús3rF   s        €r6   Úloc_mlez!rayleigh_gen.fit.<locals>.loc_mle¥#  sT   ø€ ð ˜‘ˆBØ—‘“ˆBØ�a‘%—‘“ˆBØ�B‘$—‘“ˆBØ˜˜Aœc $›i™KÑ(¨Ñ+Ñ+Ð+r8   c                 ób   •— ‰| z
  }|j                  «       |dz  d|z  j                  «       z  z
  S r‹  )r¥  )r.   r/   r#  rF   s      €r6   Úloc_mle_scale_fixedz-rayleigh_gen.fit.<locals>.loc_mle_scale_fixed®#  s2   ø€ ð ˜‘ˆBØ—6‘6“8˜e Q™h¨!¨B©$¯©«Ñ5Ñ5Ð5r8   r   Úrayleighr   rŸ  r.   r!  )r3   rA   rC   rG  rP   r£  rK  ri   r=   r•  rZ  r‡  rZ   r   r+   r[  rW   ÚflagrH  )rE   rF   rG   r5   rö   r÷   r?
  rB
  rD
  Úloc0rI   rS   rR   rõ  r.   r/   r–  s    `              €r6   rC   zrayleigh_gen.fit‘#  sF  ù€ ð �8‰8�J Ô&Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3Ü8¸¸tØ9=¸tóEÑˆˆd�Fô	Gô
	,ð ,2õ 	6ð Ðä�v‰v�d˜T‘k QÑ&Ô'Ü" :°Q¼b¿f¹fÔEÐEà™Y t›_Ð,Ð,ð �x‰x˜‹ˆØˆ<à—>‘> $Ó'¨Ñ*ˆDà˜‰gÐ-@ˆÜ—‘œbŸf™f T›l¬R¯V©V¨GÓ4ˆÜ" 3¨Ó/ˆÜ×"Ñ" 3°¸Ð0@ÔAˆØ�}Š}Ü  §¡Ó*Ð*Ø�h‰hˆØÒ(™) C›.ˆØ�EˆzÐr8   r  )rƒ   r„   r…   r†   r   r  r  rk   rÙ   rr   rÞ   ru   r~   ry   rç   r�   r   rò   rK   r	   rC   rÐ  rÑ  s   @r6   r/
  r/
  S#  su   ø„ ñð* "×4Ñ4€Mòó@ò'ò'ò&ò*ò&òò'ò'ò.ð Ù˜}ð 5.ô /ó/ó/ó ô/r8   r/
  rE
  c                   ó€   ‡ — e Zd ZdZd„ Zd„ Zˆ fd„Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ ZdZ eee¬«      ˆ fd„«       Zˆ xZS )Úreciprocal_gena,  A loguniform or reciprocal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for this class is:

    .. math::

        f(x, a, b) = \frac{1}{x \log(b/a)}

    for :math:`a \le x \le b`, :math:`b > a > 0`. This class takes
    :math:`a` and :math:`b` as shape parameters.

    %(after_notes)s

    %(example)s

    This doesn't show the equal probability of ``0.01``, ``0.1`` and
    ``1``. This is best when the x-axis is log-scaled:

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)
    >>> ax.hist(np.log10(r))
    >>> ax.set_ylabel("Frequency")
    >>> ax.set_xlabel("Value of random variable")
    >>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
    >>> ticks = ["$10^{{ {} }}$".format(i) for i in [-2, -1, 0]]
    >>> ax.set_xticklabels(ticks)  # doctest: +SKIP
    >>> plt.show()

    This random variable will be log-uniform regardless of the base chosen for
    ``a`` and ``b``. Let's specify with base ``2`` instead:

    >>> rvs = %(name)s(2**-2, 2**0).rvs(size=1000)

    Values of ``1/4``, ``1/2`` and ``1`` are equally likely with this random
    variable.  Here's the histogram:

    >>> fig, ax = plt.subplots(1, 1)
    >>> ax.hist(np.log2(rvs))
    >>> ax.set_ylabel("Frequency")
    >>> ax.set_xlabel("Value of random variable")
    >>> ax.xaxis.set_major_locator(plt.FixedLocator([-2, -1, 0]))
    >>> ticks = ["$2^{{ {} }}$".format(i) for i in [-2, -1, 0]]
    >>> ax.set_xticklabels(ticks)  # doctest: +SKIP
    >>> plt.show()

    c                 ó   — |dkD  ||kD  z  S r2  r‡   r  s      r6   rc   zreciprocal_gen._argcheck$  ó   € Ø�A‘˜!˜a™%Ñ Ð r8   c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS rg  rh   rh  s      r6   rk   zreciprocal_gen._shape_info$  rk  r8   c                 ó¶   •— t        |t        «      r|j                  «       }t        ‰| �  |t        j                  |«      t        j                  |«      f¬«      S ©NrM  ©r?   r*   r“  rA   r•  rP   r‡  r£  rT  s     €r6   r•  zreciprocal_gen._fitstart$  sC   ø€ Ü�dœLÔ)Ø—>‘>Ó#ˆDä‰wÑ  ¬R¯V©V°D«\¼2¿6¹6À$»<Ð,HÐ ÓIÐIr8   c                 ó
   — ||fS rN   r‡   r  s      r6   r–   zreciprocal_gen._get_support$  rV  r8   c                 óN   — t        j                  | j                  |||«      «      S rN   rÝ  rt  s       r6   rr   zreciprocal_gen._pdf$  rÞ  r8   c                 ó¬   — t        j                  |«       t        j                  t        j                  |«      t        j                  |«      z
  «      z
  S rN   r2  rt  s       r6   rÞ   zreciprocal_gen._logpdf$  s5   € Ü—‘�q“	ˆzœBŸF™F¤2§6¡6¨!£9¬r¯v©v°a«yÑ#8Ó9Ñ9Ð9r8   c                 ó°   — t        j                  |«      t        j                  |«      z
  t        j                  |«      t        j                  |«      z
  z  S rN   r2  rt  s       r6   ru   zreciprocal_gen._cdf$  s7   € Ü—‘�q“	œ"Ÿ&™& ›)Ñ#¬¯©¨q«	´B·F±F¸1³IÑ(=Ñ>Ð>r8   c                 ó°   — t        j                  t        j                  |«      |t        j                  |«      t        j                  |«      z
  z  z   «      S rN   ©rP   r·   rð   rƒ  s       r6   r~   zreciprocal_gen._ppf$  s8   € Ü�v‰v”b—f‘f˜Q“i !¤R§V¡V¨A£Y´·±¸³Ñ%:Ñ";Ñ;Ó<Ð<r8   c                 ó6  — |dk(  rydt        j                  |«      t        j                  |«      z
  z  |z  }t        j                  t        j                  t	        |t        j                  |«      z  |t        j                  |«      z  «      «      «      }||z  S )Nr   r‰   r   )rP   rð   rÌ  r·   Ú	_log_diff)rE   rb   r‹   rŒ   rº  r»  s         r6   r  zreciprocal_gen._munp!$  so   € Ø�Š6ØØ”"—&‘&˜“)œbŸf™f Q›iÑ'Ñ(¨1Ñ,ˆÜ�W‰W”R—V‘VœI a¬"¯&©&°«)¡m°Q´r·v±v¸a³y±[ÓAÓBÓCˆØ�B‰wˆr8   c                 óÜ   — dt        j                  |«      t        j                  |«      z   z  t        j                  t        j                  |«      t        j                  |«      z
  «      z   S r  r2  r  s      r6   rò   zreciprocal_gen._entropy($  sE   € Ø”B—F‘F˜1“I¤§¡ q£	Ñ)Ñ*¬R¯V©V´B·F±F¸1³IÄÇÁÀqÃ	Ñ4IÓ-JÑJÐJr8   z“        `loguniform`/`reciprocal` is over-parameterized. `fit` automatically
         fixes `scale` to 1 unless `fscale` is provided by the user.

ró   c                 óR   •— |j                  dd«      }t        ‰| �  |g|¢­d|i|¤ŽS )Nr÷   r   )r3   rA   rC   )rE   rF   rG   r5   r÷   r–  s        €r6   rC   zreciprocal_gen.fit/$  s1   ø€ à—‘˜( AÓ&ˆÜ‰w‰{˜4Ð> $Ò>¨vÐ>¸Ñ>Ð>r8   )rƒ   r„   r…   r†   rc   rk   r•  r–   rr   rÞ   ru   r~   r  rò   Úfit_noter	   r   rC   rÐ  rÑ  s   @r6   rI
  rI
  Ï#  s`   ø„ ñ2òf!òô
Jòò-ò:ò?ò=òòKðL€Hñ ˜}°HÔ=ó?ó >ô?r8   rI
  Ú
loguniformÚ
reciprocalc                   ó<   — e Zd ZdZd„ Zd„ Zd
d„Zd„ Zd„ Zd„ Z	d	„ Z
y)Úrice_gena  A Rice continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `rice` is:

    .. math::

        f(x, b) = x \exp(- \frac{x^2 + b^2}{2}) I_0(x b)

    for :math:`x >= 0`, :math:`b > 0`. :math:`I_0` is the modified Bessel
    function of order zero (`scipy.special.i0`).

    `rice` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    The Rice distribution describes the length, :math:`r`, of a 2-D vector with
    components :math:`(U+u, V+v)`, where :math:`U, V` are constant, :math:`u,
    v` are independent Gaussian random variables with standard deviation
    :math:`s`.  Let :math:`R = \sqrt{U^2 + V^2}`. Then the pdf of :math:`r` is
    ``rice.pdf(x, R/s, scale=s)``.

    %(example)s

    c                 ó   — |dk\  S r2  r‡   r¥	  s     r6   rc   zrice_gen._argcheck\$  rå  r8   c                 ó@   — t        dddt        j                  fd«      gS )NrŒ   Fr   rg   rh   rj   s    r6   rk   zrice_gen._shape_info_$  rè  r8   Nc                 ó®   — |t        j                  d«      z  |j                  d|z   ¬«      z   }t        j                  ||z  j                  d¬«      «      S )NrU   )rU   r  r   rú  )rP   rÿ   rÕ   r¥  )rE   rŒ   r×   rØ   rÏ  s        r6   rÙ   zrice_gen._rvsb$  sH   € àŒb�g‰g�a‹j‰L˜<×7Ñ7¸TÀD¹[Ð7ÓIÑIˆÜ�w‰w˜˜!™—y‘y a�yÓ(Ó)Ð)r8   c                 ó|   — t        j                  t        j                  |«      dt        j                  |«      «      S r  )rw   ÚchndtrrP   r"  r¼  s      r6   ru   zrice_gen._cdfg$  s%   € Ü�y‰yœŸ™ 1› q¬"¯)©)°A«,Ó7Ð7r8   c           	      ó|   — t        j                  t        j                  |dt        j                  |«      «      «      S r  )rP   rÿ   rw   Úchndtrixr"  rÉ  s      r6   r~   zrice_gen._ppfj$  s&   € Ü�w‰w”r—{‘{ 1 a¬¯©°1«Ó6Ó7Ð7r8   c                 ó~   — |t        j                  ||z
   ||z
  z  dz  «      z  t        j                  ||z  «      z  S r>  )rP   r·   rw   Úi0er¼  s      r6   rr   zrice_gen._pdfm$  s<   € ð ”2—6‘6˜A˜a™C˜& ! A¡#™, sÑ*Ó+Ñ+¬b¯f©f°Q°q±S«kÑ9Ð9r8   c                 óº   — |dz  }d|z   }||z  dz  }d|z  t        j                  | «      z  t        j                  |«      z  t        j                  |d|«      z  S r  )rP   r·   rw   rØ  Úhyp1f1)rE   rb   rŒ   Únd2Ún1rº  s         r6   r  zrice_gen._munpv$  s^   € Ø�‰eˆØ�‰WˆØˆq‰S�‰WˆØ�c‘
œRŸV™V R C›[Ñ(¬2¯8©8°B«<Ñ7Ü—	‘	˜"˜a Ó$ñ%ð 	&r8   r  )rƒ   r„   r…   r†   rc   rk   rÙ   ru   r~   rr   r  r‡   r8   r6   r^
  r^
  ?$  s+   „ ñò8òDó*ò
8ò8ò:ó&r8   r^
  Úricec                   óx   — e Zd ZdZ eed¬«      d„ «       Zd„ Zd„ Zd„ Z	d„ Z
ed	„ «       Zd
„ Zd„ Zd„ Zdd„Zd„ Zy)Úirwinhall_gena\
  An Irwin-Hall (Uniform Sum) continuous random variable.

    An `Irwin-Hall <https://en.wikipedia.org/wiki/Irwin-Hall_distribution/>`_
    continuous random variable is the sum of :math:`n` independent
    standard uniform random variables [1]_ [2]_.

    %(before_notes)s

    Notes
    -----
    Applications include `Rao's Spacing Test
    <https://jammalam.faculty.pstat.ucsb.edu/html/favorite/test.htm>`_,
    a more powerful alternative to the Rayleigh test
    when the data are not unimodal, and radar [3]_.

    Conveniently, the pdf and cdf are the :math:`n`-fold convolution of
    the ones for the standard uniform distribution, which is also the
    definition of the cardinal B-splines of degree :math:`n-1`
    having knots evenly spaced from :math:`1` to :math:`n` [4]_ [5]_.

    The Bates distribution, which represents the *mean* of statistically
    independent, uniformly distributed random variables, is simply the
    Irwin-Hall distribution scaled by :math:`1/n`. For example, the frozen
    distribution ``bates = irwinhall(10, scale=1/10)`` represents the
    distribution of the mean of 10 uniformly distributed random variables.
    
    %(after_notes)s

    References
    ----------
    .. [1] P. Hall, "The distribution of means for samples of size N drawn
            from a population in which the variate takes values between 0 and 1,
            all such values being equally probable",
            Biometrika, Volume 19, Issue 3-4, December 1927, Pages 240-244,
            :doi:`10.1093/biomet/19.3-4.240`.
    .. [2] J. O. Irwin, "On the frequency distribution of the means of samples
            from a population having any law of frequency with finite moments,
            with special reference to Pearson's Type II,
            Biometrika, Volume 19, Issue 3-4, December 1927, Pages 225-239,
            :doi:`0.1093/biomet/19.3-4.225`.
    .. [3] K. Buchanan, T. Adeyemi, C. Flores-Molina, S. Wheeland and D. Overturf, 
            "Sidelobe behavior and bandwidth characteristics
            of distributed antenna arrays,"
            2018 United States National Committee of
            URSI National Radio Science Meeting (USNC-URSI NRSM),
            Boulder, CO, USA, 2018, pp. 1-2.
            https://www.usnc-ursi-archive.org/nrsm/2018/papers/B15-9.pdf.
    .. [4] Amos Ron, "Lecture 1: Cardinal B-splines and convolution operators", p. 1
            https://pages.cs.wisc.edu/~deboor/887/lec1new.pdf.
    .. [5] Trefethen, N. (2012, July). B-splines and convolution. Chebfun. 
            Retrieved April 30, 2024, from http://www.chebfun.org/examples/approx/BSplineConv.html.

    %(example)s
    zÞ        Raises a ``NotImplementedError`` for the Irwin-Hall distribution because
        the generic `fit` implementation is unreliable and no custom implementation
        is available. Consider using `scipy.stats.fit`.

ró   c                 ó   — d}t        |«      ‚)Nz’The generic `fit` implementation is unreliable for this distribution, and no custom implementation is available. Consider using `scipy.stats.fit`.)rÜ	  )rE   rF   rG   r5   Ú	fit_notess        r6   rC   zirwinhall_gen.fit¸$  s   € ð
9ˆ	ô " )Ó,Ð,r8   c                 óP   — |dkD  t        |«      z  t        j                  |«      z  S r2  )r   rP   Ú	isrealobjra   s     r6   rc   zirwinhall_gen._argcheckÂ$  s"   € Ø�A‘œ Q›Ñ'¬"¯,©,°q«/Ñ9Ð9r8   c                 ó
   — d|fS r2  r‡   ra   s     r6   r–   zirwinhall_gen._get_supportÅ$  rV  r8   c                 ó@   — t        dddt        j                  fd«      gS rf   rh   rj   s    r6   rk   zirwinhall_gen._shape_infoÈ$  rl   r8   c                 ób   — d„ } t        j                  |t         j                  g¬«      ||«      S )Nc                 óº   — t        j                  |t         j                  ¬«      }t        j                  || z   |d¬«      t        j
                  || z   |d¬«      z  S )Nr  T)Úexact)rP   rû   Úint64rw   Ú	stirling2r¹  )r[  rb   s     r6   Úvmunpz"irwinhall_gen._munp.<locals>.vmunpÎ$  sH   € Ü—
‘
˜1¤B§H¡HÔ-ˆAÜ—L‘L  5¡¨!°4Ô8Ü—g‘g˜a ™g q°Ô5ñ6ð 7r8   r“  r•  )rE   r[  rb   rz
  s       r6   r  zirwinhall_gen._munpË$  s)   € ò	7ð 8Œr�|‰|˜E¬2¯:©:¨,Ô7¸¸qÓAÐAr8   c                 ó\   — t        j                  | dz   «      }t        j                  |«      S r^   )rP   rM  r   Úbasis_element)rb   rÏ  s     r6   Ú	_cardbsplzirwinhall_gen._cardbsplÖ$  s$   € ä�I‰I�a˜‘c‹NˆÜ×$Ñ$ QÓ'Ð'r8   c                 óh   ‡ — ˆ fd„} t        j                  |t         j                  g¬«      ||«      S )Nc                 ó2   •—  ‰j                  |«      | «      S rN   )r}
  ©rq   rb   rE   s     €r6   Úvpdfz irwinhall_gen._pdf.<locals>.vpdfÜ$  s   ø€ Ø$�4—>‘> !Ó$ QÓ'Ð'r8   r“  r•  )rE   rq   rb   r�
  s   `   r6   rr   zirwinhall_gen._pdfÛ$  s(   ø€ ô	(à6Œr�|‰|˜D¬"¯*©*¨Ô6°q¸!Ó<Ð<r8   c                 óh   ‡ — ˆ fd„} t        j                  |t         j                  g¬«      ||«      S )Nc                 óN   •—  ‰j                  |«      j                  «       | «      S rN   ©r}
  Úantiderivativer€
  s     €r6   Úvcdfz irwinhall_gen._cdf.<locals>.vcdfá$  s"   ø€ Ø5�4—>‘> !Ó$×3Ñ3Ó5°aÓ8Ð8r8   r“  r•  )rE   rq   rb   r†
  s   `   r6   ru   zirwinhall_gen._cdfà$  s(   ø€ ô	9à6Œr�|‰|˜D¬"¯*©*¨Ô6°q¸!Ó<Ð<r8   c                 óh   ‡ — ˆ fd„} t        j                  |t         j                  g¬«      ||«      S )Nc                 óT   •—  ‰j                  |«      j                  «       || z
  «      S rN   r„
  r€
  s     €r6   Úvsfzirwinhall_gen._sf.<locals>.vsfæ$  s&   ø€ Ø5�4—>‘> !Ó$×3Ñ3Ó5°a¸±cÓ:Ð:r8   r“  r•  )rE   rq   rb   r‰
  s   `   r6   ry   zirwinhall_gen._sfå$  s(   ø€ ô	;à5Œr�|‰|˜C¬¯©¨Ô5°a¸Ó;Ð;r8   Nc                 ó2   — t         dd„«       } ||||¬«      S )Nc                 ó¬   — t        j                  | «      j                  t        «      } |€| fn| g|¢­}|j	                  |¬«      j                  d¬«      S )Nr  r   rú  )rP   rG  rÃ  r  r  r¥  )rb   r×   rØ   Úusizes       r6   Ú_rvs1z!irwinhall_gen._rvs.<locals>._rvs1ë$  sO   € ä—‘˜“×"Ñ"¤3Ó'ˆAØ ˜L�Q‘D¨q¨j°4©jˆEØ×'Ñ'¨UÐ'Ó3×7Ñ7¸QÐ7Ó?Ð?r8   r×  r  )r   )rE   rb   r×   rØ   rG   r�
  s         r6   rÙ   zirwinhall_gen._rvsê$  s'   € Ü	#ò	@ó 
$ð	@ñ �Q˜T°Ô=Ð=r8   c                 ó&   — |dz  |dz  ddd|z  z  fS )NrU   rÀ  r   rÿ  rC  r‡   ra   s     r6   r   zirwinhall_gen._statsò$  s#   € ð �‰s�A�b‘D˜!˜R  1¡™XÐ%Ð%r8   r  )rƒ   r„   r…   r†   r
   r   rC   rc   r–   rk   r  rÃ  r}
  rr   ru   ry   rÙ   r   r‡   r8   r6   rn
  rn
  €$  sm   „ ñ5ñn   ð 6?ô @ñ-ó	@ð-ò:òòCò	Bð ñ(ó ð(ò=ò
=ò
<ó
>ó&r8   rn
  Ú	irwinhall)rˆ   rb   c                   ó6   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd	d„Z	y)
Úrecipinvgauss_gena­  A reciprocal inverse Gaussian continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `recipinvgauss` is:

    .. math::

        f(x, \mu) = \frac{1}{\sqrt{2\pi x}}
                    \exp\left(\frac{-(1-\mu x)^2}{2\mu^2x}\right)

    for :math:`x \ge 0`.

    `recipinvgauss` takes ``mu`` as a shape parameter for :math:`\mu`.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r‘  rh   rj   s    r6   rk   zrecipinvgauss_gen._shape_info%  r²  r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  r˜  s      r6   rr   zrecipinvgauss_gen._pdf%  s   € ô �v‰v�d—l‘l 1 bÓ)Ó*Ð*r8   c                 óJ   — t        |dkD  ||fd„ t        j                   ¬«      S )Nr   c                 óŒ   — d|| z  z
  dz   d| z  |dz  z  z  dt        j                  dt         j                  z  | z  «      z  z
  S )Nr   r¶   rU   r”   rï   )rq   rC  s     r6   rç  z+recipinvgauss_gen._logpdf.<locals>.<lambda>%%  sI   € ¨1¨r°!©t©8°c©/Ð)9¸Q¸q¹SÀÀSÁ¹[Ñ)IØ+.¬r¯v©v°a¼¿¹±g¸a±iÓ/@Ñ+@ñ*A€ r8   rÿ  r  r˜  s      r6   rÞ   zrecipinvgauss_gen._logpdf#%  s*   € Ü˜!˜a™% ! R ñBä%'§V¡V Gô-ð 	-r8   c                 óÂ   — d|z  |z
  }d|z  |z   }dt        j                  |«      z  }t        | |z  «      t        j                  d|z  «      t        | |z  «      z  z
  S ©Nr‰   r¶   ©rP   rÿ   rÀ   r·   ©rE   rq   rC  Útrm1Útrm2Úisqxs         r6   ru   zrecipinvgauss_gen._cdf)%  s_   € Ø�2‰v˜‰zˆØ�2‰v˜‰zˆØ”2—7‘7˜1“:‰~ˆÜ˜$˜˜t™Ó$¤r§v¡v¨c°"©f£~´iÀÀÀdÁ
Ó6KÑ'KÑKÐKr8   c                 óÀ   — d|z  |z
  }d|z  |z   }dt        j                  |«      z  }t        ||z  «      t        j                  d|z  «      t        | |z  «      z  z   S r—
  r˜
  r™
  s         r6   ry   zrecipinvgauss_gen._sf/%  s]   € Ø�2‰v˜‰zˆØ�2‰v˜‰zˆØ”2—7‘7˜1“:‰~ˆÜ˜˜d™Ó#¤b§f¡f¨S°©V£n´YÀ¸uÀT¹zÓ5JÑ&JÑJÐJr8   Nc                 ó0   — d|j                  |d|¬«      z  S r“  r”  r–  s       r6   rÙ   zrecipinvgauss_gen._rvs5%  s   € Ø�<×$Ñ$ R¨°4Ð$Ó8Ñ8Ð8r8   r  )
rƒ   r„   r…   r†   rk   rr   rÞ   ru   ry   rÙ   r‡   r8   r6   r‘
  r‘
  %  s(   „ ñò,Fò+ò
-òLòKô9r8   r‘
  Úrecipinvgaussc                   óB   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zdd„Z	d	„ Z
d
„ Zy)Úsemicircular_gena  A semicircular continuous random variable.

    %(before_notes)s

    See Also
    --------
    rdist

    Notes
    -----
    The probability density function for `semicircular` is:

    .. math::

        f(x) = \frac{2}{\pi} \sqrt{1-x^2}

    for :math:`-1 \le x \le 1`.

    The distribution is a special case of `rdist` with ``c = 3``.

    %(after_notes)s

    References
    ----------
    .. [1] "Wigner semicircle distribution",
           https://en.wikipedia.org/wiki/Wigner_semicircle_distribution

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zsemicircular_gen._shape_info[%  r¦   r8   c                 ó`   — dt         j                  z  t        j                  d||z  z
  «      z  S r  r  r©   s     r6   rr   zsemicircular_gen._pdf^%  s%   € Ø”2—5‘5‰yœŸ™  1 Q¡3¡›Ñ'Ð'r8   c                 óˆ   — t        j                  dt         j                  z  «      dt        j                  | |z  «      z  z   S rÂ  r  r©   s     r6   rÞ   zsemicircular_gen._logpdfa%  s0   € Ü�v‰v�aœŸ™‘g‹ ¤R§X¡X¨q¨b°©d£^Ñ!3Ñ3Ð3r8   c                 ó˜   — ddt         j                  z  |t        j                  d||z  z
  «      z  t        j                  |«      z   z  z   S )Nr”   r‰   r   )rP   rñ   rÿ   r*  r©   s     r6   ru   zsemicircular_gen._cdfd%  s<   € Ø�3”r—u‘u‘9˜a¤§¡¨¨!¨A©#©£Ñ.´·±¸1³Ñ=Ñ>Ñ>Ð>r8   c                 ó.   — t         j                  |d«      S ©Nr†  )r-
  r~   r°   s     r6   r~   zsemicircular_gen._ppfg%  s   € Ü�z‰z˜!˜QÓÐr8   Nc                 óÂ   — t        j                  |j                  |¬«      «      }t        j                  t         j                  |j                  |¬«      z  «      }||z  S r4  )rP   rÿ   r  r!  rñ   )rE   r×   rØ   rõ  r‹   s        r6   rÙ   zsemicircular_gen._rvsj%  sN   € ô �G‰G�L×(Ñ(¨dÐ(Ó3Ó4ˆÜ�F‰F”2—5‘5˜<×/Ñ/°TÐ/Ó:Ñ:Ó;ˆØ�1‰uˆr8   c                  ó   — y)N)r   rÕ  r   r<  r‡   rj   s    r6   r   zsemicircular_gen._statsq%  rm  r8   c                  ó   — y)NgzCÏ‘ ¡ä?r‡   rj   s    r6   rò   zsemicircular_gen._entropyt%  s   € Ø%r8   r  )rƒ   r„   r…   r†   rk   rr   rÞ   ru   r~   rÙ   r   rò   r‡   r8   r6   r¡
  r¡
  <%  s/   „ ñò<ò(ò4ò?ò óò ó&r8   r¡
  Úsemicircularc                   ó<   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd
d„Z	d„ Z
y	)Úskewcauchy_genaò  A skewed Cauchy random variable.

    %(before_notes)s

    See Also
    --------
    cauchy : Cauchy distribution

    Notes
    -----

    The probability density function for `skewcauchy` is:

    .. math::

        f(x) = \frac{1}{\pi \left(\frac{x^2}{\left(a\, \text{sign}(x) + 1
                                                   \right)^2} + 1 \right)}

    for a real number :math:`x` and skewness parameter :math:`-1 < a < 1`.

    When :math:`a=0`, the distribution reduces to the usual Cauchy
    distribution.

    %(after_notes)s

    References
    ----------
    .. [1] "Skewed generalized *t* distribution", Wikipedia
       https://en.wikipedia.org/wiki/Skewed_generalized_t_distribution#Skewed_Cauchy_distribution

    %(example)s

    c                 ó2   — t        j                  |«      dk  S r^   )rP   rŽ  r  s     r6   rc   zskewcauchy_gen._argcheck�%  s   € Ü�v‰v�a‹y˜1‰}Ðr8   c                 ó    — t        dddd«      gS )Nr‹   F)r<  r‰   r
  ©r   rj   s    r6   rk   zskewcauchy_gen._shape_info %  s   € Ü˜3  {°NÓCÐDÐDr8   c                 óx   — dt         j                  |dz  |t        j                  |«      z  dz   dz  z  dz   z  z  S r1  )rP   rñ   rQ   r  s      r6   rr   zskewcauchy_gen._pdf£%  s:   € Ø”B—E‘E˜Q ™T Q¬¯©°«¡^°aÑ%7¸!Ñ$;Ñ;¸aÑ?Ñ@ÑAÐAr8   c                 ó  — t        j                  |dk  d|z
  dz  d|z
  t         j                  z  t        j                  |d|z
  z  «      z  z   d|z
  dz  d|z   t         j                  z  t        j                  |d|z   z  «      z  z   «      S ©Nr   r   rU   )rP   rO  rñ   rð  r  s      r6   ru   zskewcauchy_gen._cdf¦%  s   € Ü�x‰x˜˜Q™Ø˜Q™ !™ q¨1¡u´·±¡o¼¿	¹	À!ÀqÈ1ÁuÁ+Ó8NÑ&NÑNØ˜Q™ !™ q¨1¡u´·±¡o¼¿	¹	À!ÀqÈ1ÁuÁ+Ó8NÑ&NÑNóPð 	Pr8   c           
      ó>  — || j                  d|«      k  }t        j                  |t        j                  t        j                  d|z
  z  |d|z
  dz  z
  z  «      d|z
  z  t        j                  t        j                  d|z   z  |d|z
  dz  z
  z  «      d|z   z  «      S r³
  )ru   rP   rO  r  rñ   )rE   rq   r‹   rþ  s       r6   r~   zskewcauchy_gen._ppf«%  s�   € Ø�—	‘	˜!˜Q“ÑˆÜ�x‰x˜ÜŸ™œrŸu™u¨¨A©™°!°q¸1±uÀ±k±/ÑBÓCÀqÈ1ÁuÑMÜŸ™œrŸu™u¨¨A©™°!°q¸1±uÀ±k±/ÑBÓCÀqÈ1ÁuÑMóOð 	Or8   c                 ó~   — t         j                  t         j                  t         j                  t         j                  fS rN   r›  )rE   r‹   r  s      r6   r   zskewcauchy_gen._stats±%  rœ  r8   c                 ó�   — t        |t        «      r|j                  «       }t        j                  |g d¢«      \  }}}d|||z
  dz  fS )Nr¡  rˆ   rU   r¥  )rE   rF   r¨  r©  rª  s        r6   r•  zskewcauchy_gen._fitstart´%  sE   € ô �dœLÔ)Ø—>‘>Ó#ˆDÜŸ™ dªLÓ9‰ˆˆS�#Ø�C˜# ™) Q™Ð&Ð&r8   Nr‹  )rƒ   r„   r…   r†   rc   rk   rr   ru   r~   r   r•  r‡   r8   r6   r­
  r­
  {%  s/   „ ñ òBòEòBòPò
Oó.ó'r8   r­
  Ú
skewcauchyc                   ó–   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zˆ fd„Zd„ Z	d„ Z
d	„ Zdd
„Zdd„Zed„ «       Zd„ Z eed¬«      ˆ fd„«       Zˆ xZS )Úskewnorm_genaA  A skew-normal random variable.

    %(before_notes)s

    Notes
    -----
    The pdf is::

        skewnorm.pdf(x, a) = 2 * norm.pdf(x) * norm.cdf(a*x)

    `skewnorm` takes a real number :math:`a` as a skewness parameter
    When ``a = 0`` the distribution is identical to a normal distribution
    (`norm`). `rvs` implements the method of [1]_.

    This distribution uses routines from the Boost Math C++ library for
    the computation of ``cdf``, ``ppf`` and ``isf`` methods. [2]_

    %(after_notes)s

    References
    ----------
    .. [1] A. Azzalini and A. Capitanio (1999). Statistical applications of
        the multivariate skew-normal distribution. J. Roy. Statist. Soc.,
        B 61, 579-602. :arxiv:`0911.2093`
    .. [2] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

    %(example)s

    c                 ó,   — t        j                  |«      S rN   r™  r  s     r6   rc   zskewnorm_gen._argcheckß%  rš  r8   c                 ó^   — t        ddt        j                   t        j                  fd«      gS )Nr‹   Fr
  rh   rj   s    r6   rk   zskewnorm_gen._shape_infoâ%  r�  r8   c                 ó.   — t        |dk(  ||fd„ d„ ¬«      S )Nr   c                 ó   — t        | «      S rN   rÛ   ©rq   r‹   s     r6   rç  z#skewnorm_gen._pdf.<locals>.<lambda>ç%  s
   € ¬°1«€ r8   c                 ó<   — dt        | «      z  t        || z  «      z  S r>  r
  r¾
  s     r6   rç  z#skewnorm_gen._pdf.<locals>.<lambda>è%  s   € ˜Bœy¨›|™O¬I°a¸±c«NÑ:€ r8   rê  rì  r  s      r6   rr   zskewnorm_gen._pdfå%  s"   € ÜØ�‰F�Q˜�FÑ5Ù:ô
ð 	
r8   c                 ó.   — t        |dk(  ||fd„ d„ ¬«      S )Nr   c                 ó   — t        | «      S rN   rÝ   r¾
  s     r6   rç  z&skewnorm_gen._logpdf.<locals>.<lambda>í%  s
   € ¬°a«€ r8   c                 ób   — t        j                  d«      t        | «      z   t        || z  «      z   S r  r
  r¾
  s     r6   rç  z&skewnorm_gen._logpdf.<locals>.<lambda>î%  s%   € œBŸF™F 1›I¤l°1£oÑ5´lÀ1ÀQÁ3Ó6GÑG€ r8   rê  rì  r  s      r6   rÞ   zskewnorm_gen._logpdfë%  s"   € ÜØ�‰F�Q˜�FÑ8ÙGô
ð 	
r8   c                 ó  •— t        j                  |«      }t        j                  |dd|«      }t        j                  ||j
                  «      }|dk  |dkD  z  }t        ‰| �  ||   ||   «      ||<   t        j                  |dd«      S )Nrˆ   r‰   g�íµ ÷Æ°>r   r   )	rP   rã  rn   Ú_skewnorm_cdfrm	  r·  rA   ru   r�	  )rE   rq   r‹   r�   Úi_small_cdfr–  s        €r6   ru   zskewnorm_gen._cdfñ%  s}   ø€ Ü�M‰M˜!ÓˆÜ×Ñ  3¨¨QÓ/ˆä�O‰O˜A˜sŸy™yÓ)ˆà˜T‘z a¨!¡eÑ,ˆÜ ™7™<¨¨+©¸¸+¹ÓGˆˆKÑÜ�w‰w�s˜A˜qÓ!Ð!r8   c                 ó2   — t        j                  |dd|«      S ©Nrˆ   r‰   )rn   Ú_skewnorm_ppfr  s      r6   r~   zskewnorm_gen._ppfû%  ó   € Ü× Ñ   C¨¨aÓ0Ð0r8   c                 ó*   — | j                  | | «      S rN   r–  r  s      r6   ry   zskewnorm_gen._sfþ%  s   € ð �y‰y˜!˜˜a˜RÓ Ð r8   c                 ó2   — t        j                  |dd|«      S rÇ
  )rn   Ú_skewnorm_isfr  s      r6   r�   zskewnorm_gen._isf&  rÉ
  r8   c                 ó  — |j                  |¬«      }|j                  |¬«      }|t        j                  d|dz  z   «      z  }||z  |t        j                  d|dz  z
  «      z  z   }t        j                  |dk\  || «      S )Nr  r   rU   r   )r5  rP   rÿ   rO  )rE   r‹   r×   rØ   Úu0rË  rÎ  rÚ  s           r6   rÙ   zskewnorm_gen._rvs&  s€   € Ø× Ñ  dÐ Ó+ˆØ×Ñ TÐÓ*ˆØŒb�g‰g�a˜!˜Q™$‘hÓÑˆØˆr‰T�A”b—g‘g˜a ! Q¡$™hÓ'Ñ'Ñ'ˆÜ�x‰x˜˜a™  b SÓ)Ð)r8   c                 ó¤  — g d¢}t        j                  dt         j                  z  «      |z  t        j                  d|dz  z   «      z  }d|v r||d<   d|v rd|dz  z
  |d<   d|v r;dt         j                  z
  dz  |t        j                  d|dz  z
  «      z  d	z  z  |d<   d
|v r+dt         j                  d	z
  z  |dz  d|dz  z
  dz  z  z  |d	<   |S )Nrr  rU   r   r?  r   rË  r  r$  r†  r  rW  )rE   r‹   r  r-  Úconsts        r6   r   zskewnorm_gen._stats&  sØ   € Ú)ˆÜ—‘˜œ"Ÿ%™%™Ó  1Ñ$¤R§W¡W¨Q°°A±©XÓ%6Ñ6ˆà�'‰>ØˆF�1‰IØ�'‰>Ø˜E 1™H™ˆF�1‰IØ�'‰>ØœbŸe™e™) Q™¨5´·±¸¸UÀA¹X¹Ó1FÑ+FÈÑ*JÑJˆF�1‰IØ�'‰>ØœBŸE™E A™I™¨5°!©8°Q¸À¹±\ÀAÑ4EÑ+EÑFˆF�1‰Iàˆr8   c                 óú   — t        dg«      t        ddg«      t        g d¢«      t        g d¢«      t        g d¢«      t        g d¢«      t        g d¢«      t        g d	¢«      t        g d
¢«      t        g d¢«      dœ
}|S )Nr   r†  r¿  )rÖ  iöÿÿÿr†  )éi   i—ÿÿÿé?   iñÿÿÿ)i±  iûÿÿin  iäýÿÿrÒ
  )é›(  iS¼ÿÿi6Q  iþÅÿÿi�  iOüÿÿ)iß iBàûÿi�/ iÌúÿiÉo iàþÿrÔ
  )éî iƒÔ·ÿiáç� i«Yeÿi{Hx i±óÄÿi“§ i!ðýÿ)	i!Ïi¨×…úiì‡€iø†‘ïiV ùiX'‹õiƒliˆ‘çþrÕ
  )
is_'i§áìŠl   </õ1 lýÿÿÿdy˜( l   J8²D lýÿÿÿ.~ l   ¬-Rx iìW¢i[©iß0òý)
r   r†  rC  rÉ  r  rÜ  é   rÖ  é   é   r   )rE   Úskewnorm_odd_momentss     r6   Ú_skewnorm_odd_momentsz"skewnorm_gen._skewnorm_odd_moments"&  s‚   € ô ˜1˜#‹Ü˜1˜b˜'Ó"Üš,Ó'ÜÒ.Ó/ÜÒ7Ó8ÜÒEÓFÜò #ó $äò 8ó 9äò %ó &ô ò 2ó 3ñ 
Ðð$ $Ð#r8   c                 ó  — |dz  rP|dkD  rt        d«      ‚|t        j                  d|dz  z   «      z  }| | j                  |   |dz  «      z  t        z  S t        j                  |dz   dz  «      d|dz  z  z  t        z  S )NrU   rØ
  zKskewnorm noncentral moments not implemented for odd orders greater than 19.r   )rÜ	  rP   rÿ   rÚ
  r'   rw   rØ  r&   )rE   r[  r‹   r%  s       r6   r  zskewnorm_gen._munp8&  s–   € Ø�1Š9Ø�rŠzÜ)ð +5ó 6ð 6ð
 ”b—g‘g˜a ! Q¡$™hÓ'Ñ'ˆEØÐ=˜D×6Ñ6°uÑ=¸eÀQ¹hÓGÑGÜ%ñ&ð 'ô —8‘8˜U Q™Y¨™MÓ*¨Q°°q±©\Ñ9¼HÑDÐDr8   aÕ          If ``method='mm'``, parameters fixed by the user are respected, and the
        remaining parameters are used to match distribution and sample moments
        where possible. For example, if the user fixes the location with
        ``floc``, the parameters will only match the distribution skewness and
        variance to the sample skewness and variance; no attempt will be made
        to match the means or minimize a norm of the errors.
        Note that the maximum possible skewness magnitude of a
        `scipy.stats.skewnorm` distribution is approximately 0.9952717; if the
        magnitude of the data's sample skewness exceeds this, the returned
        shape parameter ``a`` will be infinite.
        

ró   c           	      ó4  •— |j                  dd«      rt        ‰| �  |g|¢­i |¤ŽS t        |t        «      r7|j                  «       dk(  r|j                  «       }nt        ‰| �  |g|¢­i |¤ŽS t        | |||«      \  }}}}|j                  dd«      j                  «       }d„ }d„ }	|dk(  rd	\  }
}}n6t        |«      r|d   nd }
|j                  d
d «      }|j                  dd «      }|€Ä|
€Ât        j                  |«      }|dk(  rt        j                  |dd«      }n  |d«      }t        j                  || |«      } |	|«      }t        j                  d¬«      5  t        j                   t        j"                  |dz  d|dz  z
  «      «      t        j$                  |«      z  }
d d d «       n$|�|n|
}
|
t        j                   d|
dz  z   «      z  }|€J|€Ht        j&                  |«      }t        j                   |dd|dz  z  t        j(                  z  z
  z  «      }n|�|}|€G|€Et        j*                  |«      }|||z  t        j                   dt        j(                  z  «      z  z
  }n|�|}|dk(  r|
||fS t        ‰| �  ||
f||dœ|¤ŽS # 1 sw Y   ŒÄxY w)Nr;  Fr   r1   r;   c                 óÈ   — dt         j                  z
  dz  | t        j                  dt         j                  z  «      z  dz  dd| dz  z  t         j                  z  z
  dz  z  z  S )Nr$  rU   r†  r   rÊ  r  ©rÎ  s    r6   Úskew_dz skewnorm_gen.fit.<locals>.skew_di&  s]   € Ø”b—e‘e‘G˜Q‘; 1¤r§w¡w¨q´2·5±5©yÓ'9Ñ#9¸AÑ"=Ø%&¨¨1¨a©4©´"·%±%©Ñ%7¸3Ñ$?ñ#@ñ Að Ar8   c                 óê   — t        j                  | «      dz  }t        j                  | «      t        j                  t         j                  dz  |z  |dt         j                  z
  dz  dz  z   z  «      z  S )NrŠ  rU   r$  )rP   rŽ  rQ   rÿ   rñ   )rA  Ús_23s     r6   Úd_skewz skewnorm_gen.fit.<locals>.d_skewm&  s^   € Ü—6‘6˜$“< #Ñ&ˆDÜ—7‘7˜4“=¤2§7¡7Ü—‘�a‘˜$‘ $¨1¬r¯u©u©9°a©-¸3Ñ)?Ñ"?Ñ@ó$ñ ð r8   r<   rB  r.   r/   g®Gáz®ï¿g®Gáz®ï?r   r9  r:  rU   rF  )r3   rA   rC   r?   r*   r@   r“  rG  r=   r>   r¤  rò  rA  rP   r�	  r<  rÿ   r;  rQ   r§  rñ   rþ   )rE   rF   rG   r5   rš  rö   r÷   r1   rß
  râ
  r‹   r.   r/   r  Ús_maxrÎ  rË  r?  r–  s                     €r6   rC   zskewnorm_gen.fitL&  s“  ø€ ð �8‰8�J Ô&Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3Ü�dœLÔ)Ø× Ñ Ó" aÒ'Ø—~‘~Ó'‘ä‘w‘{ 4Ð7¨$Ò7°$Ñ7Ð7ô "=¸TÀ4Ø=AÀ4ó"IÑˆˆb�$˜à—‘˜( EÓ*×0Ñ0Ó2ˆò	Aò	ð �TŠ>Ø,‰MˆAˆs‘Eä˜tœ9��Q’¨$ˆAØ—(‘(˜5 $Ó'ˆCØ—H‘H˜W dÓ+ˆEàˆ:˜!˜)ô —
‘
˜4Ó ˆAØ˜Šô —G‘G˜A˜u dÓ+‘á˜q›	�Ü—G‘G˜A ˜v uÓ-�Ù�q“	ˆAÜ—‘ HÔ-ñ BÜ—G‘GœBŸI™I a¨¡d¨Q¨q°!©t©VÓ5Ó6´r·w±w¸q³zÑA�÷Bð Bð �n‘¨!ˆAØ”B—G‘G˜A  1¡™HÓ%Ñ%ˆAàˆ>˜e˜mÜ—‘�t“ˆAÜ—G‘G˜A  Q q¨!¡t¡V¬B¯E©E¡\Ñ!1Ñ2Ó3‰EØÐØˆEàˆ<˜C˜KÜ—‘˜“ˆAØ�e˜A‘gœbŸg™g a¬¯©¡gÓ.Ñ.Ñ.‰CØÐØˆCà�TŠ>Ø�c˜5�=Ð ô ‘7‘;˜t QÐE¨C°uÑEÀÑEÐE÷/Bð Bús   ÅA	JÊJr  r‹  )rƒ   r„   r…   r†   rc   rk   rr   rÞ   ru   r~   ry   r�   rÙ   r   r   rÚ
  r  r	   r   rC   rÐ  rÑ  s   @r6   r¹
  r¹
  Á%  sy   ø„ ñò:òKò
ò
ô"ò1ò!ò
1ó*óð* ñ$ó ð$ò*Eñ( ˜}ð 5ô óGFóôGFr8   r¹
  Úskewnormc                   ó:   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
y	)
Útrapezoid_gena›  A trapezoidal continuous random variable.

    %(before_notes)s

    Notes
    -----
    The trapezoidal distribution can be represented with an up-sloping line
    from ``loc`` to ``(loc + c*scale)``, then constant to ``(loc + d*scale)``
    and then downsloping from ``(loc + d*scale)`` to ``(loc+scale)``.  This
    defines the trapezoid base from ``loc`` to ``(loc+scale)`` and the flat
    top from ``c`` to ``d`` proportional to the position along the base
    with ``0 <= c <= d <= 1``.  When ``c=d``, this is equivalent to `triang`
    with the same values for `loc`, `scale` and `c`.
    The method of [1]_ is used for computing moments.

    `trapezoid` takes :math:`c` and :math:`d` as shape parameters.

    %(after_notes)s

    The standard form is in the range [0, 1] with c the mode.
    The location parameter shifts the start to `loc`.
    The scale parameter changes the width from 1 to `scale`.

    %(example)s

    References
    ----------
    .. [1] Kacker, R.N. and Lawrence, J.F. (2007). Trapezoidal and triangular
       distributions for Type B evaluation of standard uncertainty.
       Metrologia 44, 117-127. :doi:`10.1088/0026-1394/44/2/003`


    c                 ó<   — |dk\  |dk  z  |dk\  z  |dk  z  ||k\  z  S r”  r‡   ©rE   r  rÎ  s      r6   rc   ztrapezoid_gen._argcheckÇ&  s0   € Ø�Q‘˜1 ™6Ñ" a¨1¡fÑ-°°a±Ñ8¸AÀ¹FÑCÐCr8   c                 óB   — t        dddd«      }t        dddd«      }||gS )Nr  F©r   r‰   ©TTrÎ  r°
  r   s      r6   rk   ztrapezoid_gen._shape_infoÊ&  s+   € Ü˜˜U H¨lÓ;ˆÜ˜˜U H¨lÓ;ˆØ�Bˆxˆr8   c                 ój   — d||z
  dz   z  }t        ||k  ||k  ||k  z  ||kD  gd„ d„ d„ g||||f«      S )NrU   r   c                 ó   — || z  |z  S rN   r‡   ©rq   r  rÎ  r  s       r6   rç  z$trapezoid_gen._pdf.<locals>.<lambda>Õ&  s   € ¨q°1©u°q©y€ r8   c                 ó   — |S rN   r‡   rî
  s       r6   rç  z$trapezoid_gen._pdf.<locals>.<lambda>Ö&  s   € ¨q€ r8   c                 ó   — |d| z
  z  d|z
  z  S r^   r‡   rî
  s       r6   rç  z$trapezoid_gen._pdf.<locals>.<lambda>×&  s   € ¨q°A°a±C©y¸A¸a¹CÑ/@€ r8   ©r   )rE   rq   r  rÎ  r  s        r6   rr   ztrapezoid_gen._pdfÏ&  s`   € Ø��1‘�Q‘‰Kˆä˜A ™EØ !™V¨¨Q©Ñ/Ø ™Eð#ñ 9Ù0Ù@ðBð ˜q ! Q˜<ó)ð 	)r8   c                 óR   — t        ||k  ||k  ||k  z  ||kD  gd„ d„ d„ g|||f«      S )Nc                 ó$   — | dz  |z  ||z
  dz   z  S r‹  r‡   ©rq   r  rÎ  s      r6   rç  z$trapezoid_gen._cdf.<locals>.<lambda>Þ&  s   € ¨A¨q©D°1©H¸¸!¹¸A¹Ñ,>€ r8   c                 ó*   — |d| |z
  z  z   ||z
  dz   z  S r‹  r‡   rô
  s      r6   rç  z$trapezoid_gen._cdf.<locals>.<lambda>ß&  s   € ¨Q°°a¸±c±©]¸qÀ¹sÀ1¹uÑ,E€ r8   c                 ó6   — dd| z
  dz  ||z
  dz   z  d|z
  z  z
  S r1  r‡   rô
  s      r6   rç  z$trapezoid_gen._cdf.<locals>.<lambda>à&  s3   € ¨A°°!±¸©zØ23°A±#°a±%ñ09Ø<=¸a¹Cñ0Añ -B€ r8   rñ
  r3  s       r6   ru   ztrapezoid_gen._cdfÚ&  sP   € Ü˜A ™EØ !™V¨¨Q©Ñ/Ø ™Eð#ñ ?ÙEñBðCð ˜q !˜9ó&ð 	&r8   c                 óR  — | j                  |||«      | j                  |||«      }}||k  ||k  ||kD  g}t        j                  ||z  d|z   |z
  z  «      d|z  d|z   |z
  z  d|z  z   dt        j                  d|z
  ||z
  dz   z  d|z
  z  «      z
  g}t        j                  ||«      S r$  )ru   rP   rÿ   Úselect)rE   r}   r  rÎ  ÚqcÚqdrª  rx	  s           r6   r~   ztrapezoid_gen._ppfä&  s½   € Ø—‘˜1˜a Ó# T§Y¡Y¨q°!°QÓ%7ˆBˆØ˜‘F˜A ™G Q¨¡VÐ,ˆÜ—g‘g˜a !™e q¨1¡u¨q¡yÑ1Ó2Ø˜A‘g  Q¡¨¡Ñ+¨c°A©gÑ5Øœ"Ÿ'™' 1 q¡5¨Q°©U°Q©YÑ"7¸1¸q¹5Ñ"AÓBÑBðDˆ
ô �y‰y˜ :Ó.Ð.r8   c                 ó¦   ‡— |‰dz   z  }t        |dk(  d|k  |dk  z  |dk(  gd„ ˆfd„ˆfd„g|g«      }dd|z   |z
  z  ||z
  z  ‰dz   ‰dz   z  z  }|S )	Nr   rˆ   r‰   c                  ó   — yr  r‡   rÞ
  s    r6   rç  z%trapezoid_gen._munp.<locals>.<lambda>þ&  s   � r8   c                 ól   •— t        j                  ‰dz   t        j                  | «      z  «      | dz
  z  S rÏ  )rP   r  rð   ©rÎ  rb   s    €r6   rç  z%trapezoid_gen._munp.<locals>.<lambda>ÿ&  s*   ø€ ”r—x‘x  1¡¬¯©¨q«	Ñ 1Ó2°a¸±eÑ<€ r8   c                 ó   •— ‰dz   S r  r‡   rþ
  s    €r6   rç  z%trapezoid_gen._munp.<locals>.<lambda> '  s   ø€ �q˜‘s€ r8   r¶   rU   rñ
  )rE   rb   r  rÎ  Úab_termÚdc_termrx  s    `     r6   r  ztrapezoid_gen._munpì&  sƒ   ø€ ð �a˜‘c‘(ˆÜØ�#‰X˜˜a™ A¨¡GÑ,¨a°3©hÐ7ÙÛ<Ûðð ˆCóˆð �S˜‘U˜1‘W‰o ¨7Ñ!2Ñ3¸¸!¹ÀÀ!Á±}ÑEˆØˆ
r8   c                 óh   — dd|z
  |z   z  d|z   |z
  z  t        j                  dd|z   |z
  z  «      z   S r“   r2  rè
  s      r6   rò   ztrapezoid_gen._entropy'  s=   € ð �c˜!‘e˜A‘g‰ # a¡%¨¡'Ñ*¬R¯V©V°C¸3¸q¹5À¹7±OÓ-DÑDÐDr8   N)rƒ   r„   r…   r†   rc   rk   rr   ru   r~   r  rò   r‡   r8   r6   ræ
  ræ
  ¥&  s-   „ ñ òBDòò
	)ò&ò/òó2Er8   ræ
  zS`trapz` is deprecated in favour of `trapezoid` and will be removed in SciPy 1.16.0.c                   ó   — e Zd ZdZd„ Zy)Ú	trapz_genz‰

    .. deprecated:: 1.14.0
        `trapz` is deprecated and will be removed in SciPy 1.16.
        Plese use `trapezoid` instead!
    c                 óf   — t        j                  t        t        d¬«        | j                  |i |¤ŽS ©NrU   rE  )rH  rI  ÚdeprmsgÚDeprecationWarningÚfreeze)rE   rG   r5   s      r6   Ú__call__ztrapz_gen.__call__'  s)   € Ü�‰”gÔ1¸aÕ@Øˆt�{‰{˜DÐ) DÑ)Ð)r8   N)rƒ   r„   r…   r†   r
  r‡   r8   r6   r  r  '  s   „ ñó*r8   r  Ú	trapezoidÚtrapz)r�   ÚentropyÚexpectrC   Úintervalr¨  Úlogcdfré  Úlogsfrþ   r�  r]  Úpdfr¤  rÙ  rÕ  rò  Ústdr§  c                   ó   — e Zd Zd„ Zd„ Zy)Ú_DeprecationWrapperc                 óJ   — d|› d|› d�| _         t        t        |«      | _        y )Nz`trapz.z'` is deprecated in favour of trapezoid.zt. Please replace all uses of the distribution class `trapz` with `trapezoid`. `trapz` will be removed in SciPy 1.16.)rY   Úgetattrr  r1   )rE   r1   s     r6   rP  z_DeprecationWrapper.__init__/'  s1   € Ø˜f˜XÐ%LÈVÈHð UXð XˆŒô œi¨Ó0ˆ�r8   c                 ór   — t        j                  | j                  t        d¬«        | j                  |i |¤ŽS r  )rH  rI  rY   r  r1   )rE   rG   Úkwargss      r6   r
  z_DeprecationWrapper.__call__5'  s-   € Ü�‰�d—h‘hÔ 2¸qÕAØˆt�{‰{˜DÐ+ FÑ+Ð+r8   N)rƒ   r„   r…   rP  r
  r‡   r8   r6   r  r  .'  s   „ ò1ó,r8   r  c                   óB   — e Zd ZdZdd„Zd„ Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
d
„ Zy)Ú
triang_gena5  A triangular continuous random variable.

    %(before_notes)s

    Notes
    -----
    The triangular distribution can be represented with an up-sloping line from
    ``loc`` to ``(loc + c*scale)`` and then downsloping for ``(loc + c*scale)``
    to ``(loc + scale)``.

    `triang` takes ``c`` as a shape parameter for :math:`0 \le c \le 1`.

    %(after_notes)s

    The standard form is in the range [0, 1] with c the mode.
    The location parameter shifts the start to `loc`.
    The scale parameter changes the width from 1 to `scale`.

    %(example)s

    Nc                 ó*   — |j                  d|d|«      S r”  )Ú
triangularrë  s       r6   rÙ   ztriang_gen._rvsT'  s   € Ø×&Ñ& q¨!¨Q°Ó5Ð5r8   c                 ó   — |dk\  |dk  z  S r”  r‡   r~  s     r6   rc   ztriang_gen._argcheckW'  s   € Ø�Q‘˜1 ™6Ñ"Ð"r8   c                 ó    — t        dddd«      gS )Nr  Frê
  rë
  r°
  rj   s    r6   rk   ztriang_gen._shape_infoZ'  s   € Ü˜3  x°Ó>Ð?Ð?r8   c                 ó`   — t        |dk(  ||k  ||k\  |dk7  z  |dk(  gd„ d„ d„ d„ g||f«      }|S )Nr   r   c                 ó   — dd| z  z
  S r  r‡   r£  s     r6   rç  z!triang_gen._pdf.<locals>.<lambda>g'  s   €  a¨!¨a©%¡i€ r8   c                 ó   — d| z  |z  S r  r‡   r£  s     r6   rç  z!triang_gen._pdf.<locals>.<lambda>h'  ó   €  a¨!¡e¨a¡i€ r8   c                 ó   — dd| z
  z  d|z
  z  S r‹  r‡   r£  s     r6   rç  z!triang_gen._pdf.<locals>.<lambda>i'  s   €  a¨1¨q©5¡k°Q¸±UÑ&;€ r8   c                 ó   — d| z  S r  r‡   r£  s     r6   rç  z!triang_gen._pdf.<locals>.<lambda>j'  ó
   €  a¨!¡e€ r8   rñ
  ©rE   rq   r  rõ  s       r6   rr   ztriang_gen._pdf]'  sZ   € ô ˜˜a™Ø˜Q™Ø˜q™& Q¨!¡VÑ,Ø˜a™ð!ñ 0Ù/Ù;Ù+ð-ð ˜A˜ó ˆð ˆr8   c                 ó`   — t        |dk(  ||k  ||k\  |dk7  z  |dk(  gd„ d„ d„ d„ g||f«      }|S )Nr   r   c                 ó   — d| z  | | z  z
  S r  r‡   r£  s     r6   rç  z!triang_gen._cdf.<locals>.<lambda>s'  s   €  a¨¡c¨A¨a©C¡i€ r8   c                 ó   — | | z  |z  S rN   r‡   r£  s     r6   rç  z!triang_gen._cdf.<locals>.<lambda>t'  r#  r8   c                 ó*   — | | z  d| z  z
  |z   |dz
  z  S r‹  r‡   r£  s     r6   rç  z!triang_gen._cdf.<locals>.<lambda>u'  s   €  q¨¡s¨Q¨q©S¡y°1¡}¸¸1¹Ñ&=€ r8   c                 ó   — | | z  S rN   r‡   r£  s     r6   rç  z!triang_gen._cdf.<locals>.<lambda>v'  r&  r8   rñ
  r'  s       r6   ru   ztriang_gen._cdfn'  sX   € Ü˜˜a™Ø˜Q™Ø˜q™& Q¨!¡VÑ,Ø˜a™ð!ñ 0Ù/Ù=Ù+ð-ð ˜A˜ó ˆð ˆr8   c           
      ó    — t        j                  ||k  t        j                  ||z  «      dt        j                  d|z
  d|z
  z  «      z
  «      S r^   )rP   rO  rÿ   r  s      r6   r~   ztriang_gen._ppfz'  s?   € Ü�x‰x˜˜A™œrŸw™w q¨1¡u›~¨q´·±¸!¸A¹#À!ÀAÁ#¹Ó1GÑ/GÓHÐHr8   c           	      óÈ   — |dz   dz  d|z
  ||z  z   dz  t        j                  d«      d|z  dz
  z  |dz   z  |dz
  z  dt        j                  d|z
  ||z  z   d«      z  z  dfS )	Nr‰   rD  é   rU   r   rC  rÊ  g333333ã¿)rP   rÿ   rÌ  r~  s     r6   r   ztriang_gen._stats}'  sx   € Ø�3‘˜‘Ø�Q‘�q˜‘s‘˜B‘Ü—‘˜“
˜A˜a™C ™EÑ" A a¡CÑ(¨!¨A©#Ñ.°!´B·H±H¸cÀ!¹eÀAÀaÁC¹iÈ#Ó4NÑ2NÑOØðð 	r8   c                 ó2   — dt        j                  d«      z
  S r»  r2  r~  s     r6   rò   ztriang_gen._entropyƒ'  s   € Ø”2—6‘6˜!“9‰}Ðr8   r  )rƒ   r„   r…   r†   rÙ   rc   rk   rr   ru   r~   r   rò   r‡   r8   r6   r  r  >'  s1   „ ñó*6ò#ò@òò"
òIòór8   r  Útriangc                   óX   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zˆ fd
„Zd„ Zˆ xZS )Útruncexpon_genad  A truncated exponential continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `truncexpon` is:

    .. math::

        f(x, b) = \frac{\exp(-x)}{1 - \exp(-b)}

    for :math:`0 <= x <= b`.

    `truncexpon` takes ``b`` as a shape parameter for :math:`b`.

    %(after_notes)s

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS rº  rh   rj   s    r6   rk   ztruncexpon_gen._shape_info '  r  r8   c                 ó   — | j                   |fS rN   r4  r¥	  s     r6   r–   ztruncexpon_gen._get_support£'  ó   € Ø�v‰v�qˆyÐr8   c                 ó^   — t        j                  | «      t        j                  | «       z  S rN   rÃ  r¼  s      r6   rr   ztruncexpon_gen._pdf¦'  s#   € ä�v‰v�q�b‹zœBŸH™H a R›L˜=Ñ)Ð)r8   c                 ó^   — | t        j                  t        j                  | «       «      z
  S rN   rÈ  r¼  s      r6   rÞ   ztruncexpon_gen._logpdfª'  s$   € Øˆr”B—F‘FœBŸH™H a R›L˜=Ó)Ñ)Ð)r8   c                 ó\   — t        j                  | «      t        j                  | «      z  S rN   ry  r¼  s      r6   ru   ztruncexpon_gen._cdf­'  s!   € Ü�x‰x˜˜‹|œBŸH™H a R›LÑ(Ð(r8   c                 ó\   — t        j                  |t        j                  | «      z  «       S rN   )rw   r¦  r  rÉ  s      r6   r~   ztruncexpon_gen._ppf°'  s"   € Ü—‘˜œ2Ÿ8™8 Q B›<™Ó(Ð(Ð(r8   c                 óŠ   — t        j                  | «      t        j                  | «      z
  t        j                  | «      z  S rN   rÃ  r¼  s      r6   ry   ztruncexpon_gen._sf³'  s0   € Ü—‘˜�r“
œRŸV™V Q B›ZÑ'¬¯©°1°"«Ñ5Ð5r8   c                 óŠ   — t        j                  t        j                  | «      |t        j                  | «      z  z
  «       S rN   )rP   rð   r·   rw   r  rÉ  s      r6   r�   ztruncexpon_gen._isf¶'  s2   € Ü—‘”r—v‘v˜q˜b“z A¬¯©°!°«Ñ$4Ñ4Ó5Ð5Ð5r8   c                 ó2  •— |dk(  r7d|dz   t        j                  | «      z  z
  t        j                  | «       z  S |dk(  rFddd||z  d|z  z   dz   z  t        j                  | «      z  z
  z  t        j                  | «       z  S t        ‰| �  ||«      S rI  )rP   r·   rw   r  rA   r  )rE   rb   rŒ   r–  s      €r6   r  ztruncexpon_gen._munp¹'  s™   ø€ ð �Š6Ø�q˜‘sœBŸF™F A 2›JÑ&Ñ&¬"¯(©(°A°2«,¨Ñ7Ð7Ø�!ŠVØ�a˜˜Q˜q™S  1¡™W Q™Y™¬¯©°¨r«
Ñ2Ñ2Ñ3´b·h±hÀ¸r³l°]ÑCÐCô ‘7‘=  AÓ&Ð&r8   c                 ó€   — t        j                  |«      }t        j                  |dz
  «      d||dz
  z  z   d|z
  z  z   S r¿  rU
  )rE   rŒ   ÚeBs      r6   rò   ztruncexpon_gen._entropyÄ'  s;   € Ü�V‰V�A‹YˆÜ�v‰v�b˜‘d‹|˜Q˜r 1 S¡5™z™\¨C°©FÑ3Ñ3Ð3r8   )rƒ   r„   r…   r†   rk   r–   rr   rÞ   ru   r~   ry   r�   r  rò   rÐ  rÑ  s   @r6   r3  r3  Š'  s;   ø„ ñò*Eòò*ò*ò)ò)ò6ò6ô	'ö4r8   r3  Ú
truncexpon)rˆ   rŒ   c                 ó4   — t        j                  | |gd¬«      S )Nr   rú  )rw   rE  ©Úlog_pÚlog_qs     r6   Ú_log_sumrE  Î'  s   € Ü�<‰<˜ ˜¨QÔ/Ð/r8   c                 ó\   — t        j                  | |t        j                  dz  z   gd¬«      S )Nù              ð?r   rú  )rw   rE  rP   rñ   rB  s     r6   rW
  rW
  Ó'  s$   € Ü�<‰<˜ ¤b§e¡e¨B¡h¡Ð/°aÔ8Ð8r8   c                 óÄ  ‡— t        j                  | |«      \  } }|dk  }| dkD  }||z   }d„ Šˆfd„}d„ }t        j                  | t         j                  t         j                  ¬«      }| |   j
                  r ‰| |   ||   «      ||<   | |   j
                  r || |   ||   «      ||<   | |   j
                  r || |   ||   «      ||<   t        j                  |«      S )z3Log of Gaussian probability mass within an intervalr   c                 ó>   — t        t        |«      t        | «      «      S rN   )rW
  rÃ   r®  s     r6   Úmass_case_leftz'_log_gauss_mass.<locals>.mass_case_leftá'  s   € Üœ a›¬,°q«/Ó:Ð:r8   c                 ó   •—  ‰| |  «      S rN   r‡   )r‹   rŒ   rJ  s     €r6   Úmass_case_rightz(_log_gauss_mass.<locals>.mass_case_rightä'  s   ø€ Ù˜q˜b 1 "Ó%Ð%r8   c                 óZ   — t        j                  t        | «       t        | «      z
  «      S rN   )rw   r¦  rÀ   r®  s     r6   Úmass_case_centralz*_log_gauss_mass.<locals>.mass_case_centralç'  s$   € ô �x‰xœ 1›˜¬	°1°"«Ñ5Ó6Ð6r8   )r–  r	  )rP   rñ  r  r  Ú
complex128r×   rÌ  )	r‹   rŒ   Ú	case_leftÚ
case_rightÚcase_centralrL  rN  rö  rJ  s	           @r6   Ú_log_gauss_massrS  ×'  sã   ø€ ä×Ñ˜q !Ó$�D€A€qð �Q‘€IØ�Q‘€JØ Ñ+Ð,€Lò;ô&ò7ô �,‰,�q¤R§V¡V´2·=±=Ô
A€CØˆ�|×ÒÙ'¨¨)©°a¸	±lÓCˆˆI‰Øˆ�}×ÒÙ)¨!¨J©-¸¸:¹ÓGˆˆJ‰Øˆ�×ÒÙ-¨a°©o¸qÀ¹ÓOˆˆLÑÜ�7‰7�3‹<Ðr8   c                   óx   ‡ — e Zd ZdZd„ Zd„ Zˆ fd„Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zd„ Zd„ Zd„ Zd„ Zdd„Zˆ xZS )Útruncnorm_genaw
  A truncated normal continuous random variable.

    %(before_notes)s

    Notes
    -----
    This distribution is the normal distribution centered on ``loc`` (default
    0), with standard deviation ``scale`` (default 1), and truncated at ``a``
    and ``b`` *standard deviations* from ``loc``. For arbitrary ``loc`` and
    ``scale``, ``a`` and ``b`` are *not* the abscissae at which the shifted
    and scaled distribution is truncated.

    .. note::
        If ``a_trunc`` and ``b_trunc`` are the abscissae at which we wish
        to truncate the distribution (as opposed to the number of standard
        deviations from ``loc``), then we can calculate the distribution
        parameters ``a`` and ``b`` as follows::

            a, b = (a_trunc - loc) / scale, (b_trunc - loc) / scale

        This is a common point of confusion. For additional clarification,
        please see the example below.

    %(example)s

    In the examples above, ``loc=0`` and ``scale=1``, so the plot is truncated
    at ``a`` on the left and ``b`` on the right. However, suppose we were to
    produce the same histogram with ``loc = 1`` and ``scale=0.5``.

    >>> loc, scale = 1, 0.5
    >>> rv = truncnorm(a, b, loc=loc, scale=scale)
    >>> x = np.linspace(truncnorm.ppf(0.01, a, b),
    ...                 truncnorm.ppf(0.99, a, b), 100)
    >>> r = rv.rvs(size=1000)

    >>> fig, ax = plt.subplots(1, 1)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
    >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
    >>> ax.set_xlim(a, b)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    Note that the distribution is no longer appears to be truncated at
    abscissae ``a`` and ``b``. That is because the *standard* normal
    distribution is first truncated at ``a`` and ``b``, *then* the resulting
    distribution is scaled by ``scale`` and shifted by ``loc``. If we instead
    want the shifted and scaled distribution to be truncated at ``a`` and
    ``b``, we need to transform these values before passing them as the
    distribution parameters.

    >>> a_transformed, b_transformed = (a - loc) / scale, (b - loc) / scale
    >>> rv = truncnorm(a_transformed, b_transformed, loc=loc, scale=scale)
    >>> x = np.linspace(truncnorm.ppf(0.01, a, b),
    ...                 truncnorm.ppf(0.99, a, b), 100)
    >>> r = rv.rvs(size=10000)

    >>> fig, ax = plt.subplots(1, 1)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
    >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
    >>> ax.set_xlim(a-0.1, b+0.1)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()
    c                 ó   — ||k  S rN   r‡   r  s      r6   rc   ztruncnorm_gen._argcheck@(  s   € Ø�1‰uˆr8   c                 ó¾   — t        ddt        j                   t        j                  fd«      }t        ddt        j                   t        j                  fd«      }||gS )Nr‹   Frg   rŒ   )FTrh   rh  s      r6   rk   ztruncnorm_gen._shape_infoC(  sI   € Ü˜˜U¤b§f¡f W¬b¯f©fÐ$5°}ÓEˆÜ˜˜U¤b§f¡f W¬b¯f©fÐ$5°}ÓEˆØ�Bˆxˆr8   c                 ó¶   •— t        |t        «      r|j                  «       }t        ‰| �  |t        j                  |«      t        j                  |«      f¬«      S rN
  rO
  rT  s     €r6   r•  ztruncnorm_gen._fitstartH(  sC   ø€ ä�dœLÔ)Ø—>‘>Ó#ˆDÜ‰wÑ  ¬R¯V©V°D«\¼2¿6¹6À$»<Ð,HÐ ÓIÐIr8   c                 ó
   — ||fS rN   r‡   r  s      r6   r–   ztruncnorm_gen._get_supportN(  rV  r8   c                 óN   — t        j                  | j                  |||«      «      S rN   rÝ  rt  s       r6   rr   ztruncnorm_gen._pdfQ(  rY  r8   c                 ó2   — t        |«      t        ||«      z
  S rN   )r½   rS  rt  s       r6   rÞ   ztruncnorm_gen._logpdfT(  s   € Ü˜A‹¤°°AÓ!6Ñ6Ð6r8   c                 óN   — t        j                  | j                  |||«      «      S rN   rO  rt  s       r6   ru   ztruncnorm_gen._cdfW(  rY  r8   c           
      óT  — t        j                  |||«      \  }}}t        j                  t        ||«      t        ||«      z
  «      }|dkD  }t        j                  |«      rFt        j
                  t        j                  | j                  ||   ||   ||   «      «       «      ||<   |S ©Ngš™™™™™¹¿)rP   rñ  rû   rS  r£  r¦  r·   rç   )rE   rq   r‹   rŒ   r  rþ  s         r6   rã   ztruncnorm_gen._logcdfZ(  s�   € Ü×%Ñ% a¨¨AÓ.‰ˆˆ1ˆaÜ—‘œO¨A¨qÓ1´OÀAÀqÓ4IÑIÓJˆØ�T‰MˆÜ�6‰6�!Œ9ÜŸ™¤"§&¡&¨¯©°Q°q±T¸1¸Q¹4ÀÀ1ÁÓ)FÓ"GÐ!GÓHˆF�1‰IØˆr8   c                 óN   — t        j                  | j                  |||«      «      S rN   r6  rt  s       r6   ry   ztruncnorm_gen._sfb(  r7  r8   c           
      óT  — t        j                  |||«      \  }}}t        j                  t        ||«      t        ||«      z
  «      }|dkD  }t        j                  |«      rFt        j
                  t        j                  | j                  ||   ||   ||   «      «       «      ||<   |S r^  )rP   rñ  rû   rS  r£  r¦  r·   rã   )rE   rq   r‹   rŒ   r  rþ  s         r6   rç   ztruncnorm_gen._logsfe(  s�   € Ü×%Ñ% a¨¨AÓ.‰ˆˆ1ˆaÜ—
‘
œ?¨1¨aÓ0´?À1ÀaÓ3HÑHÓIˆØ�D‰LˆÜ�6‰6�!Œ9Ü—x‘x¤§¡¨¯©°Q°q±T¸1¸Q¹4ÀÀ1ÁÓ(FÓ!GÐ GÓHˆE�!‰HØˆr8   c                 ó&  — t        |«      }t        |«      }||z
  }t        j                  t        j                  dt        j                  z  t        j
                  z  «      |z  «      }|t        |«      z  |t        |«      z  z
  d|z  z  }||z   }|S r  )rÀ   rP   rð   rÿ   rñ   rm  rº   )	rE   r‹   rŒ   rm  rn  r7  rC  ÚDró  s	            r6   rò   ztruncnorm_gen._entropym(  s}   € Ü�a‹LˆÜ�a‹LˆØ�‰EˆÜ�F‰F”2—7‘7˜1œrŸu™u™9¤r§t¡tÑ+Ó,¨qÑ0Ó1ˆØ”˜1“Ñ ¤I¨a£LÑ 0Ñ0°Q¸±UÑ;ˆØ�‰EˆØˆr8   c                 ó  — t        j                  |||«      \  }}}|dk  }| }d„ }d„ }t        j                  |«      }||   }	||   }
|	j                  r ||	||   ||   «      ||<   |
j                  r ||
||   ||   «      ||<   |S )Nr   c                 ó–   — t        t        |«      t        j                  | «      t	        ||«      z   «      }t        j                  |«      S rN   )rE  rÃ   rP   rð   rS  rw   Ú	ndtri_exp©r}   r‹   rŒ   Ú	log_Phi_xs       r6   Úppf_leftz$truncnorm_gen._ppf.<locals>.ppf_left|(  s9   € Ü ¤¨a£Ü!#§¡¨£¬_¸QÀÓ-BÑ!BóDˆIä—<‘< 	Ó*Ð*r8   c                 óœ   — t        t        | «      t        j                  |  «      t	        ||«      z   «      }t        j                  |«       S rN   )rE  rÃ   rP   r¦  rS  rw   re  rf  s       r6   Ú	ppf_rightz%truncnorm_gen._ppf.<locals>.ppf_right�(  sA   € Ü ¤¨q¨bÓ!1Ü!#§¡¨1¨"£´ÀÀ1Ó0EÑ!EóGˆIä—L‘L Ó+Ð+Ð+r8   ©rP   rñ  Ú
empty_liker×   )rE   r}   r‹   rŒ   rP  rQ  rh  rj  rö  Úq_leftÚq_rights              r6   r~   ztruncnorm_gen._ppfv(  sŸ   € Ü×%Ñ% a¨¨AÓ.‰ˆˆ1ˆaà˜‘Eˆ	Ø�Zˆ
ò	+ò
	,ô
 �m‰m˜AÓˆà�9‘ˆØ�J‘-ˆà�;Š;Ù% f¨a°	©l¸A¸i¹LÓIˆC�	‰NØ�<Š<Ù'¨°°:±ÀÀ*ÁÓNˆC�
‰Oàˆ
r8   c                 ó  — t        j                  |||«      \  }}}|dk  }| }d„ }d„ }t        j                  |«      }||   }	||   }
|	j                  r ||	||   ||   «      ||<   |
j                  r ||
||   ||   «      ||<   |S )Nr   c                 ó¼   — t        t        |«      t        j                  | «      t	        ||«      z   «      }t        j                  t        j                  |«      «      S rN   )rW
  rÃ   rP   rð   rS  rw   re  rÌ  rf  s       r6   Úisf_leftz$truncnorm_gen._isf.<locals>.isf_left™(  sB   € Ü!¤,¨q£/Ü"$§&¡&¨£)¬o¸aÀÓ.CÑ"CóEˆIä—<‘<¤§¡¨	Ó 2Ó3Ð3r8   c                 óÂ   — t        t        | «      t        j                  |  «      t	        ||«      z   «      }t        j                  t        j                  |«      «       S rN   )rW
  rÃ   rP   r¦  rS  rw   re  rÌ  rf  s       r6   Ú	isf_rightz%truncnorm_gen._isf.<locals>.isf_rightž(  sJ   € Ü!¤,°¨rÓ"2Ü"$§(¡(¨A¨2£,´ÀÀAÓ1FÑ"FóHˆIä—L‘L¤§¡¨Ó!3Ó4Ð4Ð4r8   rk  )rE   r}   r‹   rŒ   rP  rQ  rq  rs  rö  rm  rn  s              r6   r�   ztruncnorm_gen._isf’(  sŸ   € ä×%Ñ% a¨¨AÓ.‰ˆˆ1ˆaà˜‘Eˆ	Ø�Zˆ
ò	4ò
	5ô
 �m‰m˜AÓˆà�9‘ˆØ�J‘-ˆà�;Š;Ù% f¨a°	©l¸A¸i¹LÓIˆC�	‰NØ�<Š<Ù'¨°°:±ÀÀ*ÁÓNˆC�
‰Oàˆ
r8   c                 ó²   ‡ — ˆ fd„}t        |dk\  ||k(  z  ||k(  z  |||ft        j                  |t        j                  g¬«      t        j                  «      S )Nc                 ó0  •‡	— ‰
j                  t        j                  ||g«      ||«      \  }}|| g}ddg}t        d| dz   «      D ]J  Š	t	        ||||ggˆ	fd„d¬«      }t        j
                  |«      ‰	dz
  |d   z  z   }|j                  |«       ŒL |d   S )zƒ
            Returns n-th moment. Defined only if n >= 0.
            Function cannot broadcast due to the loop over n
            r   r   c                 ó   •— | |‰dz
  z  z  S r^   r‡   )rq   r�  r  s     €r6   rç  z:truncnorm_gen._munp.<locals>.n_th_moment.<locals>.<lambda>¾(  s   ø€ ¨q°1°q¸±s±8©|€ r8   rÿ  r  r¿  )rr   rP   rû   rý  r   r¥  r  )rb   r‹   rŒ   ÚpAÚpBÚprobsr  r  Úmkr  rE   s            @€r6   Ún_th_momentz(truncnorm_gen._munp.<locals>.n_th_moment°(  sª   ù€ ð
 —Y‘YœrŸz™z¨1¨a¨&Ó1°1°aÓ8‰FˆB�Ø˜"˜�IˆEØ˜!�fˆGÜ˜1˜a ™c“]ò #�ô
 " %¨%°!°Q°¨Û";ÀqôJ�ä—V‘V˜D“\ Q q¡S¨G°B©KÑ$7Ñ7�Ø—‘˜rÕ"ð#ð ˜2‘;Ðr8   r   r“  rU  )rE   rb   r‹   rŒ   r{  s   `    r6   r  ztruncnorm_gen._munp¯(  sR   ø€ ô	ô& ˜1 ™6 a¨1¡fÑ-°°a±Ñ8¸1¸aÀ¸)ÜŸ,™, {¼B¿J¹J¸<ÔHÜŸ&™&ó"ð 	"r8   c                 ó¤   — | j                  t        j                  ||g«      ||«      \  }}d„ }t        j                  |d¬«      } ||||||«      S )Nc                 óH  — ||z
  }|}|| g}t        ||| |ggd„ d¬«      }dt        j                  |«      z   }	t        ||| |z
  ||z
  ggd„ d¬«      }dt        j                  |«      z   }
t        ||| |ggd„ d¬«      }d|z  t        j                  |«      z   }t        ||| |ggd„ d¬«      }d	|	z  t        j                  |«      z   }||d
|	z  d|dz  z  z   z  z   }|t        j                  |
d«      z  }||d|z  d	|z  d|	z  |dz  z
  z  z   z  z   }||
dz  z  d	z
  }||
||fS )Nc                 ó   — | |z  S rN   r‡   r£  s     r6   rç  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>Ï(  s
   € À1ÀQÁ3€ r8   r   rÿ  r   c                 ó   — | |z  S rN   r‡   r£  s     r6   rç  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>Ò(  s
   € ÈÈ1É€ r8   c                 ó   — | |dz  z  S r  r‡   r£  s     r6   rç  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>×(  ó   € À1ÀQÈÁTÁ6€ r8   rU   c                 ó   — | |dz  z  S r§
  r‡   r£  s     r6   rç  zGtruncnorm_gen._stats.<locals>._truncnorm_stats_scalar.<locals>.<lambda>Ú(  r�  r8   r†  rÔ  rÊ  rÕ  )r   rP   r¥  rÌ  )r‹   rŒ   rw  rx  r  r=  rC  ry  r  r>  rD  r?  Úm4Úmu3rE  Úmu4rF  s                    r6   Ú_truncnorm_stats_scalarz5truncnorm_gen._stats.<locals>._truncnorm_stats_scalarÊ(  sg  € Ø�b‘ˆBØˆBà˜"˜�IˆEÜ˜e e¨a°¨V _Ñ6FØ()ô+ˆDà”R—V‘V˜D“\Ñ!ˆBÜ˜e e¨a°©d°A°b±D¨\Ð%:Ñ<LØ()ô+ˆDð ”b—f‘f˜T“lÑ"ˆCÜ˜e e¨a°¨V _Ñ6IØ()ô+ˆDà�2‘œŸ™˜t›Ñ$ˆBÜ˜e e¨a°¨V _Ñ6IØ()ô+ˆDà�2‘œŸ™˜t›Ñ$ˆBà�r˜R ™U Q r¨1¡u¡W™_Ñ-Ñ-ˆCØ”r—x‘x  SÓ)Ñ)ˆBØ�r˜2˜b™5 1 R¡4¨¨2©°°A±©Ñ#6Ñ6Ñ7Ñ7ˆCØ�s˜A‘v‘ Ñ!ˆBØ�s˜B �?Ð"r8   )r  )Úexcluded)r  rP   r›  r�  )rE   r‹   rŒ   r  rw  rx  r†  Ú_truncnorm_statss           r6   r   ztruncnorm_gen._statsÇ(  sT   € Ø—‘œ"Ÿ(™( A q 6Ó*¨A¨qÓ1‰ˆˆBò	#ô4 Ÿ<™<Ð(?Ø1=ô?Ðá  1 b¨"¨gÓ6Ð6r8   r  )rƒ   r„   r…   r†   rc   rk   r•  r–   rr   rÞ   ru   rã   ry   rç   rò   r~   r�   r  r   rÐ  rÑ  s   @r6   rU  rU  ÿ'  sU   ø„ ñ>ò@òô
Jòò-ò7ò-òò,òòòò8ò:"÷07r8   rU  Ú	truncnorm)r�   r¢   c                   ó–   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zd„ Zd„ Zd„ Zd„ Zd„ Ze ee«      ˆ fd„«       «       Zˆ xZS )Útruncpareto_genac  An upper truncated Pareto continuous random variable.

    %(before_notes)s

    See Also
    --------
    pareto : Pareto distribution

    Notes
    -----
    The probability density function for `truncpareto` is:

    .. math::

        f(x, b, c) = \frac{b}{1 - c^{-b}} \frac{1}{x^{b+1}}

    for :math:`b > 0`, :math:`c > 1` and :math:`1 \le x \le c`.

    `truncpareto` takes `b` and `c` as shape parameters for :math:`b` and
    :math:`c`.

    Notice that the upper truncation value :math:`c` is defined in
    standardized form so that random values of an unscaled, unshifted variable
    are within the range ``[1, c]``.
    If ``u_r`` is the upper bound to a scaled and/or shifted variable,
    then ``c = (u_r - loc) / scale``. In other words, the support of the
    distribution becomes ``(scale + loc) <= x <= (c*scale + loc)`` when
    `scale` and/or `loc` are provided.

    %(after_notes)s

    References
    ----------
    .. [1] Burroughs, S. M., and Tebbens S. F.
        "Upper-truncated power laws in natural systems."
        Pure and Applied Geophysics 158.4 (2001): 741-757.

    %(example)s

    c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS )NrŒ   Frˆ   r
  r  r‰   rh   )rE   rj  r!  s      r6   rk   ztruncpareto_gen._shape_info)  s;   € Ü˜˜U S¬"¯&©& M°>ÓBˆÜ˜˜U S¬"¯&©& M°>ÓBˆØ�Bˆxˆr8   c                 ó   — |dkD  |dkD  z  S rÇ
  r‡   ©rE   rŒ   r  s      r6   rc   ztruncpareto_gen._argcheck)  s   € Ø�B‘˜1˜r™6Ñ"Ð"r8   c                 ó   — | j                   |fS rN   r4  rŽ  s      r6   r–   ztruncpareto_gen._get_support)  r6  r8   c                 ó2   — |||dz    z  z  dd||z  z  z
  z  S r^   r‡   ©rE   rq   rŒ   r  s       r6   rr   ztruncpareto_gen._pdf")  s'   € Ø�1˜˜!™�f‘9‰}  A a¨¡d¡F¡
Ñ+Ð+r8   c           	      óæ   — t        j                  |«      t        j                  t        j                  | t        j                  |«      z  «       «      z
  |dz   t        j                  |«      z  z
  S r^   )rP   rð   r  r‘  s       r6   rÞ   ztruncpareto_gen._logpdf%)  sM   € Ü�v‰v�a‹yœ2Ÿ6™6¤2§8¡8¨Q¨B¬r¯v©v°a«y©LÓ#9Ð"9Ó:Ñ:¸aÀ¹cÄ2Ç6Á6È!Ã9¹_ÑLÐLr8   c                 ó,   — d|| z  z
  dd||z  z  z
  z  S r^   r‡   r‘  s       r6   ru   ztruncpareto_gen._cdf()  s#   € Ø�A˜�r‘E‘	˜a ! A q¡D¡&™jÑ)Ð)r8   c                 ón   — t        j                  || z   «      t        j                  d||z  z  «      z
  S r<  r9  r‘  s       r6   rã   ztruncpareto_gen._logcdf+)  s/   € Ü�x‰x˜˜Q˜B™˜Ó¤"§(¡(¨2¨a°©d©7Ó"3Ñ3Ð3r8   c                 ó>   — t        ddd||z  z  z
  |z  z
  d|z  «      S ©Nr   r¿  r4  ©rE   r}   rŒ   r  s       r6   r~   ztruncpareto_gen._ppf.)  s(   € Ü�1˜˜A˜a ™d™F™
 A‘~Ñ% r¨!¡tÓ,Ð,r8   c                 ó8   — || z  d||z  z  z
  dd||z  z  z
  z  S r^   r‡   r‘  s       r6   ry   ztruncpareto_gen._sf1)  s+   € Ø�A�2‘˜˜!˜Q™$™‘ 1 q¨¨A©¡v¡:Ñ.Ð.r8   c                 ó~   — t        j                  || z  d||z  z  z
  «      t        j                  d||z  z  «      z
  S r–  rå  r‘  s       r6   rç   ztruncpareto_gen._logsf4)  s9   € Ü�v‰v�a˜!˜‘e˜a  1¡™f‘nÓ%¬¯©°°A°q±D±Ó(9Ñ9Ð9r8   c                 óJ   — t        d||z  z  dd||z  z  z
  |z  z   d|z  «      S r–  r4  r—  s       r6   r�   ztruncpareto_gen._isf7)  s0   € Ü�1�Q˜‘T‘6˜Q  1 a¡4¡™Z¨™NÑ*¨B¨q©DÓ1Ð1r8   c                 óœ   — t        j                  |dd||z  z  z
  z  «      |dz   t        j                  |«      ||z  dz
  z  d|z  z
  z  z    S r^   r2  rŽ  s      r6   rò   ztruncpareto_gen._entropy:)  sW   € Ü—‘˜˜1˜q  A¡™v™:™Ó'Ø�a‘Cœ"Ÿ&™& ›) Q¨¡T¨A¡XÑ.°°1±Ñ4Ñ5ñ6ð 7ð 	7r8   c                 óª   — ||k(  j                  «       r$|t        j                  |«      z  dd||z  z  z
  z  S |||z
  z  ||z  ||z  z
  z  ||z  dz
  z  S r^   )rý   rP   rð   )rE   rb   rŒ   r  s       r6   r  ztruncpareto_gen._munp>)  s^   € Ø�‰F�<‰<Œ>Ø”R—V‘V˜A“Y‘; ! a¨¨1©¡f¡*Ñ-Ð-à˜˜!™‘9  1¡ q¨!¡t¡Ñ,°°1±°q±Ñ9Ð9r8   c                 ó¢   — t        |t        «      r|j                  «       }t        j	                  |«      \  }}}t        |«      |z
  |z  }||||fS rN   )r?   r*   r“  r¨	  rC   r£  )rE   rF   rŒ   r.   r/   r  s         r6   r•  ztruncpareto_gen._fitstartD)  sM   € Ü�dœLÔ)Ø—>‘>Ó#ˆDÜŸ
™
 4Ó(‰ˆˆ3�Ü�‹Y˜‰_˜eÑ#ˆØ�!�S˜%ÐÐr8   c                 óî  •‡ ‡‡‡ ‡!‡"‡#‡$‡%— |j                  dd«      rt        ‰&‰ �  ‰g|¢­i |¤ŽS d„ Š#d„ Š"ˆˆ"ˆ#fd„Šˆ%fd„Š ˆ$ˆ%fd„}ˆ$fd„Š!dˆˆˆ ˆ!ˆ"fd	„	}d
„ }ˆ&ˆ fd„}t        ‰ ‰||«      }|\  Š}	}
}}‰j	                  «       ‰j                  «       cŠ$Š%t        j                  ‰$t        j                   «      }|	�|
�|�|�t        d«      ‚|
�€ø|�€õ|�€ò|	�€ˆˆ ˆ!ˆ"fd„}t        j                  ‰$t        j                   «      }|}d}|dz
  }|t        j                   kD  r[ ||«       ||«      z  dk\  rG|dz  }|t        j                  d|«      z
  }|t        j                   kD  r ||«       ||«      z  dk\  rŒG|t        j                   kD  s |‰g|¢­i |¤ŽS t        |||f¬«      }|j                  s |‰g|¢­i |¤ŽS |j                  dz
  }|dz
  }d}|t        j                   kD  r[ ||«       ||«      z  dk\  rG|dz  }|t        j                  d|«      z
  }|t        j                   kD  r ||«       ||«      z  dk\  rŒG|t        j                   kD  s |‰g|¢­i |¤ŽS t        |||f¬«      }|j                  s |‰g|¢­i |¤ŽS |j                  } ‰!|«      } ‰ ||«      } ‰|||«      }‰|z
  |z  }t	        d ‰#|«      z  d ‰"|«      dz
  z  «      }||k  �sý |‰g|¢­i |¤ŽS |}|dz
  }d}|t        j                   kD  rN |||	«       |||	«      z  dk\  r8|dz  }|d|z  z
  }|t        j                   kD  r |||	«       |||	«      z  dk\  rŒ8|t        j                   kD  s |‰g|¢­i |¤ŽS t        ||	f||f¬«      }|j                  s |‰g|¢­i |¤ŽS |j                  } ‰!|«      } ‰ ||«      }|	}�n|�|n ||
|«      }|xs  ‰!|«      }|
xs	  ‰ ||«      }|�$‰j	                  «       |z
  dk  rt        dd|¬«      ‚|
r2|�0|r.‰j                  «       |
|z  |z   kD  rt        dd ‰ ||«      ¬«      ‚|	€˜‰|z
  |z  } ‰#|«      }t        j                  |«      }d|z  |k  s |‰g|¢­i |¤ŽS d|z  d||z
  z  z   }t        j                  d|z  d«      }	 t        |||f||f¬«      }|j                  s |‰g|¢­i |¤ŽS |j                  }n|	}||z   ‰$k  sF|r&t        j                  |t        j                   «      }n ‰!|«      }t        j                  |d«      }||z  |z   ‰%kD  s- ‰ ||«      }t        j                  |t        j                  «      }t        j                   ‰ j#                  ||«      «      r|dkD  s |‰g|¢­i |¤ŽS ||||f}|€9|€7 |‰g|¢­i |¤Ž}‰ j%                  |‰«      }‰ j%                  |‰«      }||k  r|S |S # t        $ r |}Y �Œw xY w)Nr;  Fc                 óR   — t        j                  t        j                  | «      «      S rN   )rP   rþ   rð   r¹   s    r6   Úlog_meanz%truncpareto_gen.fit.<locals>.log_meanQ)  s   € Ü—7‘7œ2Ÿ6™6 !›9Ó%Ð%r8   c                 ó8   — dt        j                  d| z  «      z  S r^   )rP   rþ   r¹   s    r6   Ú	harm_meanz&truncpareto_gen.fit.<locals>.harm_meanT)  s   € Ø”R—W‘W˜Q˜q™S“\‘>Ð!r8   c                 ó¤   •— ‰|z
  |z  } ‰|«      } ‰	|«      }|dz
  |z  }d|dz
  |dd| z  z
  |z  t        j                  | «      z  z
  z  z
  |z  S r^   r2  )
r  r.   r/   r  Úharm_mÚlog_mÚquotrF   r¢  r   s
          €€€r6   Úget_bz"truncpareto_gen.fit.<locals>.get_bW)  si   ø€ Ø�c‘˜5Ñ ˆAÙ˜q“\ˆFÙ˜Q“KˆEØ˜1‘H˜eÑ#ˆDØ˜˜a™ D¨A°°!±©G°VÑ+;¼B¿F¹FÀ1»IÑ+EÑ$EÑFÑFÈÑMÐMr8   c                 ó   •— ‰| z
  |z  S rN   r‡   )r.   r/   Úmxs     €r6   Úget_cz"truncpareto_gen.fit.<locals>.get_c^)  s   ø€ Ø˜‘H˜eÑ#Ð#r8   c                 ó<   •— |r‰|z
  }|S | r| ‰z  ‰z
  | dz
  z  }|S y r^   r‡   )rI  r÷   r.   r¯  r©  s      €€r6   Úget_locz$truncpareto_gen.fit.<locals>.get_loca)  s7   ø€ ÙØ˜6‘k�Ø�
ÙØ˜"‘u˜r‘z B¨¡FÑ+�Ø�
ð r8   c                 ó   •— ‰| z
  S rN   r‡   )r.   r¯  s    €r6   rï	  z&truncpareto_gen.fit.<locals>.get_scalei)  s   ø€ Ø˜‘8ˆOr8   c                 óª   •—  ‰	| «      } ‰| |«      }|€
 ‰|| |«      n|} ‰
‰| z
  |z  «      }dd|dz
  ||dz   z  |z
  z  z   dd|dz   z  z
  z  |z  z
  S r^   r‡   )r.   ræ  r/   r  rŒ   r¤  rF   r§  rª  rï	  r¢  s         €€€€€r6   rT  z$truncpareto_gen.fit.<locals>.dL_dLoco)  s|   ø€ ñ ˜c“NˆEÙ�c˜5Ó!ˆAØ(*¨
‘�a˜˜eÔ$¸ˆAÙ  s¡
¨EÑ1Ó2ˆFØ˜˜Q ™U Q¨¨1©¡X°¡\Ñ2Ñ2°q¸1¸aÀ¹c¹7±{ÑCÀfÑLÑLÐLr8   c                 óP   — | t        j                  | |z  d| |z  z
  z  «      |z  z
  S r^   r9  )rŒ   ÚlogcÚlogms      r6   ÚdL_dBz"truncpareto_gen.fit.<locals>.dL_dBx)  s.   € ð ”r—x‘x  $¡¨!¨a°©f©*Ñ 5Ó6¸Ñ=Ñ=Ð=r8   c                 ó2   •— t        t        ‰�
  | g|¢­i |¤ŽS rN   )rA   r‹  rC   )rF   rG   r  r–  rE   s      €€r6   Úfallbackz%truncpareto_gen.fit.<locals>.fallback~)  s   ø€ äœ¨$Ñ3°DÐJ¸4ÒJÀ6ÑJÐJr8   z2All parameters fixed.There is nothing to optimize.c                 ó�   •—  ‰| «      } ‰| |«      } ‰‰| z
  |z  «      }dd|dz
  z  z   t        j                  |«      z  |z  dz
  S r^   r2  )r.   r/   r  r¤  rF   rª  rï	  r¢  s       €€€€r6   Úcond_bz#truncpareto_gen.fit.<locals>.cond_b�)  sT   ø€ á% c›N�EÙ˜c 5Ó)�AÙ&¨¨s©
°EÑ'9Ó:�FØ  1 Q¡3¡™K¬2¯6©6°!«9Ñ4°vÑ=ÀÑAÐAr8   r   r   r¶   r!  gü©ñÒMbP?rU   ÚtruncparetorŸ  rN   )r3   rA   rC   rG  r‡  r£  rP   rZ  ri   rú   rÌ  r+   r[  rH  rK  rð   rý   rc   rV  )'rE   rF   rG   r5   r¬  rT  r²  r´  r\  r�  rI  rö   r÷   Úmn_infr¶  rS   rþ  rR   rõ  r.   r/   r  rŒ   Ústd_dataÚ
up_bound_br±  r°  Úparams_overrideÚparams_superÚnllf_overrideÚ
nllf_superr§  rª  rï	  r¢  r   r¯  r©  r–  s'   ``                             @@@@@@@€r6   rC   ztruncpareto_gen.fitK)  s¨  ÿù€ ð �8‰8�J Ô&Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ò	&ò	"ö	Nô	$õ	ô	÷	Mñ 	Mò	>õ	Kô 1°°t¸TÀ4ÓHˆ
Ø%/Ñ"ˆˆb�"�d˜FØ—‘“˜TŸX™X›ZˆˆˆBÜ—‘˜b¤2§6¡6 'Ó*ˆàˆNØ�NØÐ$ØÐ&Üð =ó >ð >à‰Z˜D™L¨V©^Ø‰z÷Bô Ÿ™ b¬2¯6©6¨'Ó2�Ø�Ø�Ø !™�Ø¤"§&¡& Ò(Ù" 6›N©6°&«>Ñ9¸QÒ>Ø˜‘F�AØ#¤b§h¡h¨r°1£oÑ5�Fð ¤"§&¡& Ò(Ù" 6›N©6°&«>Ñ9¸QÓ>ð ¤§¡ Ò'Ù# DÐ8¨4Ò8°4Ñ8Ð8Ü! &°6¸6Ð2BÔC�Ø—}’}Ù# DÐ8¨4Ò8°4Ñ8Ð8ð Ÿ™ D™�Ø !™�Ø�Ø¤"§&¡& Ò(Ù# F›O©G°F«OÑ;¸qÒ@Ø˜‘F�AØ#¤b§h¡h¨r°1£oÑ5�Fð ¤"§&¡& Ò(Ù# F›O©G°F«OÑ;¸qÓ@ð ¤§¡ Ò'Ù# DÐ8¨4Ò8°4Ñ8Ð8Ü! '°F¸FÐ3CÔD�Ø—}’}Ù# DÐ8¨4Ò8°4Ñ8Ð8Ø—h‘h�Ù! #›�Ù˜#˜uÓ%�Ù˜!˜S %Ó(�à  3™J¨Ñ-�ä  ¡8¨HÓ#5Ñ!5Ø!"¡I¨hÓ$7¸Ñ$9Ñ!:ó<�
à˜J›Ù# DÐ8¨4Ò8°4Ñ8Ð8ð  �Ø !™�Ø�à¤§¡ Ò'Ù# F¨BÓ/Ù% f¨bÓ1ñ2Ø56ò7à˜‘F�AØ# a¨¡d™]�Fð	 ¤§¡ Ò'Ù# F¨BÓ/Ù% f¨bÓ1ñ2Ø56ó7ð ¤§¡ Ò'Ù# DÐ8¨4Ò8°4Ñ8Ð8Ü! '¨B¨5Ø+1°6Ð*:ô<�à—}’}Ù# DÐ8¨4Ò8°4Ñ8Ð8Ø—h‘h�Ù! #›�Ù˜#˜uÓ%�Ø’ð Ð*‘$±¸¸FÓ0CˆCØÒ,™i¨›nˆEØÒ'‘e˜C Ó'ˆAð Ð D§H¡H£J°Ñ$5¸Ò$9Ü" =¸ÀÔCÐCñ �tÐ'©VØ—8‘8“:  6¡	¨DÑ 0Ò0Ü& }¸AÙ-2°3¸Ó->ô@ð @ð ˆzØ  3™J¨Ñ-�Ù Ó)�Ü—v‘v˜a“y�à˜$™ šÙ# DÐ8¨4Ò8°4Ñ8Ð8à˜4™ ! T¨D¡[¡/Ñ1�ÜŸ™ a¨¡f¨aÓ0�ðÜ% e¨d°D¨\Ø/5°vÐ.>ô@�Cð Ÿ=š=Ù'¨Ð<¨tÒ<°tÑ<Ð<ØŸ™‘Að �ð �c‘	˜RÒÙÜ—l‘l 3¬¯©¨Ó0‘á! #›�ÜŸ™ U¨AÓ.�Ø�%‘˜‘˜rÒ!Ù�c˜5Ó!ˆAÜ—‘˜Q¤§¡Ó'ˆAä—‘�t—~‘~ a¨Ó+Ô,°%¸!²)Ù˜DÐ0 4Ò0¨4Ñ0Ð0à˜Q  UÐ*ˆØˆ<˜F˜Nñ
 $ DÐ8¨4Ò8°4Ñ8ˆLØ ŸI™I o°tÓ<ˆMØŸ™ <°Ó6ˆJØ˜MÒ)Ø#Ð#àÐøôE "ò Ø“Aðús   Ò-)W% ÓW% ×%W4×3W4)rƒ   r„   r…   r†   rk   rc   r–   rr   rÞ   ru   rã   r~   ry   rç   r�   rò   r  r•  rK   r   r   rC   rÐ  rÑ  s   @r6   r‹  r‹  í(  ss   ø„ ñ'òRò
#òò,òMò*ò4ò-ò/ò:ò2ò7ò:ò ð Ù˜MÓ*óZó +ó ôZr8   r‹  r·  )r‰   r  c                   óX   — e Zd ZdZej
                  Zd„ Zd„ Zd„ Z	d„ Z
d„ Zd„ Zd„ Zd	„ Zy
)Útukeylambda_gena*  A Tukey-Lamdba continuous random variable.

    %(before_notes)s

    Notes
    -----
    A flexible distribution, able to represent and interpolate between the
    following distributions:

    - Cauchy                (:math:`lambda = -1`)
    - logistic              (:math:`lambda = 0`)
    - approx Normal         (:math:`lambda = 0.14`)
    - uniform from -1 to 1  (:math:`lambda = 1`)

    `tukeylambda` takes a real number :math:`lambda` (denoted ``lam``
    in the implementation) as a shape parameter.

    %(after_notes)s

    %(example)s

    c                 ó,   — t        j                  |«      S rN   r™  ©rE   Úlams     r6   rc   ztukeylambda_gen._argcheckG*  s   € Ü�{‰{˜3ÓÐr8   c                 ó^   — t        ddt        j                   t        j                  fd«      gS )NrÃ  Fr
  rh   rj   s    r6   rk   ztukeylambda_gen._shape_infoJ*  s%   € Ü˜5 %¬2¯6©6¨'´2·6±6Ð):¸NÓKÐLÐLr8   c                 óP   — t        |dkD  |fd„ t        j                  ¬«      }| |fS )Nr   c                 ó   — d| z  S r^   r‡   )rÃ  s    r6   rç  z.tukeylambda_gen._get_support.<locals>.<lambda>O*  s
   €  Q s¡U€ r8   rù  r  )rE   rÃ  rŒ   s      r6   r–   ztukeylambda_gen._get_supportM*  s-   € Ü�s˜Q‘w  Ù*Ü!#§¡ô)ˆð ˆr�1ˆuˆr8   c           	      ó¨  — t        j                  t        j                  ||«      «      }||dz
  z  t        j                  d|z
  «      |dz
  z  z   }t        j                  d¬«      5  dt        j                  |«      z  }t        j
                  |dk  t        |«      dt        j                  |«      z  k  z  |d«      cd d d «       S # 1 sw Y   y xY w)Nr‰   r   r9  r:  r   rˆ   )rP   rû   rw   Útklmbdar<  rO  rŽ  )rE   rq   rÃ  ÚFxr]  s        r6   rr   ztukeylambda_gen._pdfS*  s©   € Ü�Z‰ZœŸ
™
 1 cÓ*Ó+ˆØ�#�c‘'‰]œbŸj™j¨¨2©Ó.°#°c±'Ñ:Ñ:ˆÜ�[‰[ Ô)ñ 	RØ”R—Z‘Z “^Ñ#ˆBÜ—8‘8˜S A™X¬#¨a«&°3´r·z±zÀ#³Ñ3FÑ*FÑGÈÈSÓQ÷	R÷ 	Rò 	Rús   Á'ACÃCc                 ó.   — t        j                  ||«      S rN   )rw   rÈ  )rE   rq   rÃ  s      r6   ru   ztukeylambda_gen._cdfZ*  s   € Ü�z‰z˜!˜SÓ!Ð!r8   c                 ó^   — t        j                  ||«      t        j                  | |«      z
  S rN   )rw   r­  r«  )rE   r}   rÃ  s      r6   r~   ztukeylambda_gen._ppf]*  s%   € Ü�y‰y˜˜CÓ ¤2§;¡;°¨r°3Ó#7Ñ7Ð7r8   c                 ó2   — dt        |«      dt        |«      fS r2  )Ú_tlvarÚ_tlkurtrÂ  s     r6   r   ztukeylambda_gen._stats`*  s   € Ø”&˜“+˜q¤'¨#£,Ð.Ð.r8   c                 óB   ‡— ˆfd„}t        j                  |dd«      d   S )Nc                 ón   •— t        j                  t        | ‰dz
  «      t        d| z
  ‰dz
  «      z   «      S r^   )rP   rð   rÈ  )rô  rÃ  s    €r6   Úintegz'tukeylambda_gen._entropy.<locals>.integd*  s/   ø€ Ü—6‘6œ#˜a  Q¡›-¬¨A¨a©C°°Q±«Ñ7Ó8Ð8r8   r   r   )r   r¡  )rE   rÃ  rÑ  s    ` r6   rò   ztukeylambda_gen._entropyc*  s    ø€ ô	9ä�~‰~˜e Q¨Ó*¨1Ñ-Ð-r8   N)rƒ   r„   r…   r†   r   r  r  rc   rk   r–   rr   ru   r~   r   rò   r‡   r8   r6   rÀ  rÀ  .*  s>   „ ñð, "×4Ñ4€Mò òMòòRò"ò8ò/ó.r8   rÀ  Útukeylambdac                   ó   — e Zd Zd„ Zy)ÚFitUniformFixedScaleDataErrorc                 ó    — d|› d|› d�| _         y )Nz Invalid values in `data`.  Maximum likelihood estimation with the uniform distribution and fixed scale requires that np.ptp(data) <= fscale, but np.ptp(data) = z and fscale = r2   rM  )rE   rý	  r÷   s      r6   rP  z&FitUniformFixedScaleDataError.__init__m*  s$   € ð:à:=¸ð ?Ø�x˜qð"ð 	�	r8   N)rƒ   r„   r…   rP  r‡   r8   r6   rÔ  rÔ  l*  s   „ ó
r8   rÔ  c                   óL   — e Zd ZdZd„ Zdd„Zd„ Zd„ Zd„ Zd„ Z	d	„ Z
ed
„ «       Zy)Úuniform_gena  A uniform continuous random variable.

    In the standard form, the distribution is uniform on ``[0, 1]``. Using
    the parameters ``loc`` and ``scale``, one obtains the uniform distribution
    on ``[loc, loc + scale]``.

    %(before_notes)s

    %(example)s

    c                 ó   — g S rN   r‡   rj   s    r6   rk   zuniform_gen._shape_info‚*  r¦   r8   Nc                 ó(   — |j                  dd|«      S rÇ
  )r  rÖ   s      r6   rÙ   zuniform_gen._rvs…*  s   € Ø×#Ñ# C¨¨dÓ3Ð3r8   c                 ó   — d||k(  z  S r  r‡   r©   s     r6   rr   zuniform_gen._pdfˆ*  s   € Ø�A˜‘F‰|Ðr8   c                 ó   — |S rN   r‡   r©   s     r6   ru   zuniform_gen._cdf‹*  ó   € Øˆr8   c                 ó   — |S rN   r‡   r°   s     r6   r~   zuniform_gen._ppfŽ*  rÜ  r8   c                  ó   — y)N)r”   gUUUUUUµ?r   g333333ó¿r‡   rj   s    r6   r   zuniform_gen._stats‘*  s   € Ø#r8   c                  ó   — yrO  r‡   rj   s    r6   rò   zuniform_gen._entropy”*  rƒ  r8   c                 ó¬  — t        |«      dkD  rt        d«      ‚|j                  dd«      }|j                  dd«      }t        |«       |�|�t	        d«      ‚t        j                  |«      }t        j                  |«      j                  «       st	        d«      ‚|€a|€&|j                  «       }t        j                  |«      }n{|}|j                  «       |z
  }|j                  «       |k  rSt        d|||z   ¬	«      ‚t        j                  |«      }||kD  rt        ||¬
«      ‚|j                  «       d||z
  z  z
  }|}t        |«      t        |«      fS )a–	  
        Maximum likelihood estimate for the location and scale parameters.

        `uniform.fit` uses only the following parameters.  Because exact
        formulas are used, the parameters related to optimization that are
        available in the `fit` method of other distributions are ignored
        here.  The only positional argument accepted is `data`.

        Parameters
        ----------
        data : array_like
            Data to use in calculating the maximum likelihood estimate.
        floc : float, optional
            Hold the location parameter fixed to the specified value.
        fscale : float, optional
            Hold the scale parameter fixed to the specified value.

        Returns
        -------
        loc, scale : float
            Maximum likelihood estimates for the location and scale.

        Notes
        -----
        An error is raised if `floc` is given and any values in `data` are
        less than `floc`, or if `fscale` is given and `fscale` is less
        than ``data.max() - data.min()``.  An error is also raised if both
        `floc` and `fscale` are given.

        Examples
        --------
        >>> import numpy as np
        >>> from scipy.stats import uniform

        We'll fit the uniform distribution to `x`:

        >>> x = np.array([2, 2.5, 3.1, 9.5, 13.0])

        For a uniform distribution MLE, the location is the minimum of the
        data, and the scale is the maximum minus the minimum.

        >>> loc, scale = uniform.fit(x)
        >>> loc
        2.0
        >>> scale
        11.0

        If we know the data comes from a uniform distribution where the support
        starts at 0, we can use ``floc=0``:

        >>> loc, scale = uniform.fit(x, floc=0)
        >>> loc
        0.0
        >>> scale
        13.0

        Alternatively, if we know the length of the support is 12, we can use
        ``fscale=12``:

        >>> loc, scale = uniform.fit(x, fscale=12)
        >>> loc
        1.5
        >>> scale
        12.0

        In that last example, the support interval is [1.5, 13.5].  This
        solution is not unique.  For example, the distribution with ``loc=2``
        and ``scale=12`` has the same likelihood as the one above.  When
        `fscale` is given and it is larger than ``data.max() - data.min()``,
        the parameters returned by the `fit` method center the support over
        the interval ``[data.min(), data.max()]``.

        r   r…  rö   Nr÷   rø   rù   r  rŸ  )rý	  r÷   r”   )r¤  r4   r3   r7   rú   rP   rû   rü   rý   r‡  rý	  r£  rK  rÔ  rR  )	rE   rF   rG   r5   rö   r÷   r.   r/   rý	  s	            r6   rC   zuniform_gen.fit—*  sF  € ôV ˆt‹9�qŠ=ÜÐ1Ó2Ð2à�x‰x˜ Ó%ˆØ—‘˜( DÓ)ˆä$ TÔ*àÐ Ð 2äð )ó *ð *ô �z‰z˜$Óˆä�{‰{˜4Ó ×$Ñ$Ô&ÜÐCÓDÐDð> ˆ>àˆ|à—h‘h“j�ÜŸ™˜t›‘ð �ØŸ™›
 SÑ(�Ø—8‘8“: Ò#Ü& y¸À3ÈÁ;ÔOÐOô —&‘&˜“,ˆCØ�VŠ|Ü3¸ÀFÔKÐKð —(‘(“*˜s F¨S¡LÑ1Ñ1ˆCØˆEô �S‹zœ5 ›<Ð'Ð'r8   r  )rƒ   r„   r…   r†   rk   rÙ   rr   ru   r~   r   rò   rK   rC   r‡   r8   r6   r×  r×  v*  s@   „ ñ
òó4òòòò$òð ñR(ó ñR(r8   r×  r  c                   óÂ   ‡ — e Zd ZdZd„ Zd„ Zdd„Z ee«      ˆ fd„«       Z	d„ Z
d„ Zd„ Zd	„ Zd
„ Z eed¬«      	 	 dˆ fd„	«       Ze eed¬«      ˆ fd„«       «       Zˆ xZS )Úvonmises_genaU  A Von Mises continuous random variable.

    %(before_notes)s

    See Also
    --------
    scipy.stats.vonmises_fisher : Von-Mises Fisher distribution on a
                                  hypersphere

    Notes
    -----
    The probability density function for `vonmises` and `vonmises_line` is:

    .. math::

        f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I_0(\kappa) }

    for :math:`-\pi \le x \le \pi`, :math:`\kappa \ge 0`. :math:`I_0` is the
    modified Bessel function of order zero (`scipy.special.i0`).

    `vonmises` is a circular distribution which does not restrict the
    distribution to a fixed interval. Currently, there is no circular
    distribution framework in SciPy. The ``cdf`` is implemented such that
    ``cdf(x + 2*np.pi) == cdf(x) + 1``.

    `vonmises_line` is the same distribution, defined on :math:`[-\pi, \pi]`
    on the real line. This is a regular (i.e. non-circular) distribution.

    Note about distribution parameters: `vonmises` and `vonmises_line` take
    ``kappa`` as a shape parameter (concentration) and ``loc`` as the location
    (circular mean). A ``scale`` parameter is accepted but does not have any
    effect.

    Examples
    --------
    Import the necessary modules.

    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy.stats import vonmises

    Define distribution parameters.

    >>> loc = 0.5 * np.pi  # circular mean
    >>> kappa = 1  # concentration

    Compute the probability density at ``x=0`` via the ``pdf`` method.

    >>> vonmises.pdf(0, loc=loc, kappa=kappa)
    0.12570826359722018

    Verify that the percentile function ``ppf`` inverts the cumulative
    distribution function ``cdf`` up to floating point accuracy.

    >>> x = 1
    >>> cdf_value = vonmises.cdf(x, loc=loc, kappa=kappa)
    >>> ppf_value = vonmises.ppf(cdf_value, loc=loc, kappa=kappa)
    >>> x, cdf_value, ppf_value
    (1, 0.31489339900904967, 1.0000000000000004)

    Draw 1000 random variates by calling the ``rvs`` method.

    >>> sample_size = 1000
    >>> sample = vonmises(loc=loc, kappa=kappa).rvs(sample_size)

    Plot the von Mises density on a Cartesian and polar grid to emphasize
    that it is a circular distribution.

    >>> fig = plt.figure(figsize=(12, 6))
    >>> left = plt.subplot(121)
    >>> right = plt.subplot(122, projection='polar')
    >>> x = np.linspace(-np.pi, np.pi, 500)
    >>> vonmises_pdf = vonmises.pdf(x, loc=loc, kappa=kappa)
    >>> ticks = [0, 0.15, 0.3]

    The left image contains the Cartesian plot.

    >>> left.plot(x, vonmises_pdf)
    >>> left.set_yticks(ticks)
    >>> number_of_bins = int(np.sqrt(sample_size))
    >>> left.hist(sample, density=True, bins=number_of_bins)
    >>> left.set_title("Cartesian plot")
    >>> left.set_xlim(-np.pi, np.pi)
    >>> left.grid(True)

    The right image contains the polar plot.

    >>> right.plot(x, vonmises_pdf, label="PDF")
    >>> right.set_yticks(ticks)
    >>> right.hist(sample, density=True, bins=number_of_bins,
    ...            label="Histogram")
    >>> right.set_title("Polar plot")
    >>> right.legend(bbox_to_anchor=(0.15, 1.06))

    c                 ó@   — t        dddt        j                  fd«      gS )Nr¡  Fr   rg   rh   rj   s    r6   rk   zvonmises_gen._shape_info�+  s   € Ü˜7 E¨A¬r¯v©v¨;¸ÓFÐGÐGr8   c                 ó   — |dk\  S r2  r‡   r±  s     r6   rc   zvonmises_gen._argcheck“+  s   € Ø˜‰zÐr8   c                 ó*   — |j                  d||¬«      S )Nrˆ   r  )Úvonmises)rE   r¡  r×   rØ   s       r6   rÙ   zvonmises_gen._rvs–+  s   € Ø×$Ñ$ S¨%°dÐ$Ó;Ð;r8   c                 ó´   •— t        ‰| �  |i |¤Ž}t        j                  |t        j                  z   dt        j                  z  «      t        j                  z
  S r  )rA   rÙ  rP   Úmodrñ   )rE   rG   r5   rÙ  r–  s       €r6   rÙ  zvonmises_gen.rvs™+  s@   ø€ ä‰g‰k˜4Ð( 4Ñ(ˆÜ�v‰v�cœBŸE™E‘k 1¤R§U¡U¡7Ó+¬b¯e©eÑ3Ð3r8   c                 ó¬   — t        j                  |t        j                  |«      z  «      dt         j                  z  t        j
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  r£  s      r6   rr   zvonmises_gen._pdfž+  s:   € ô
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                  |«      «      z
  S r  )rw   rê  rP   rð   rñ   rg
  r£  s      r6   rÞ   zvonmises_gen._logpdf¥+  s@   € à”r—x‘x “{Ñ"¤R§V¡V¨A¬b¯e©e©G£_Ñ4´r·v±v¼b¿f¹fÀU»mÓ7LÑLÐLr8   c                 ó.   — t        j                  ||«      S rN   )r   Úvon_mises_cdfr£  s      r6   ru   zvonmises_gen._cdf©+  s   € Ü×#Ñ# E¨1Ó-Ð-r8   c                  ó   — yrl  r‡   r±  s     r6   Ú_stats_skipzvonmises_gen._stats_skip¬+  rm  r8   c                 óà   — | t        j                  |«      z  t        j                  |«      z  t        j                  dt        j
                  z  t        j                  |«      z  «      z   |z   S r  )rw   Úi1erg
  rP   rð   rñ   r±  s     r6   rò   zvonmises_gen._entropy¯+  sV   € ð �œŸ™ ›Ñ&¬¯©°«Ñ6Ü—‘�qœ2Ÿ5™5‘y¤2§6¡6¨%£=Ñ0Ó1ñ2Ø49ñ:ð 	;r8   z¢        The default limits of integration are endpoints of the interval
        of width ``2*pi`` centered at `loc` (e.g. ``[-pi, pi]`` when
        ``loc=0``).

ró   c           	      óŽ   •— t         j                   t         j                  }
}	|€||	z   }|€||
z   }t        ‰| �  |||||||fi |¤ŽS rN   )rP   rñ   rA   r  )rE   r^  rG   r.   r/   ÚlbÚubÚconditionalr5   rÚ  rÙ  r–  s              €r6   r  zvonmises_gen.expect»+  s_   ø€ ô —%‘%�œŸ™ˆBˆàˆ:Ø�r‘ˆBØˆ:Ø�r‘ˆBä‰w‰~˜d D¨#Ø# R¨¨[ñBØ<@ñBð 	Br8   a          Fit data is assumed to represent angles and will be wrapped onto the
        unit circle. `f0` and `fscale` are ignored; the returned shape is
        always the maximum likelihood estimate and the scale is always
        1. Initial guesses are ignored.

c                 óþ  •— |j                  dd«      rt        ‰| �  |g|¢­i |¤ŽS t        | |||«      \  }}}}| j                  t
        j                   k(  rt        ‰| �  |g|¢­i |¤ŽS t        j                  |dt
        j                  z  «      }d„ }d„ }|�|n ||«      }	|�|n |||	«      }
t        j                  |	t
        j                  z   dt
        j                  z  «      t
        j                  z
  }	|
|	dfS )Nr;  FrU   c                 ó,   — t        j                  | «      S rN   )rò  Úcircmean)rF   s    r6   Úfind_muz!vonmises_gen.fit.<locals>.find_muÞ+  s   € Ü—>‘> $Ó'Ð'r8   c                 ój  ‡— t        j                  t        j                  || z
  «      «      t        | «      z  Š‰dk(  ry‰dkD  rNˆfd„}‰d‰z
  z  d‰z   z  }d|z  } ||«      dk\  r|S  ||«      dk  r|S t	        |d||f¬«      }|j
                  S t        j                  t        «      j                  S )Nr   g €à7yÃACr   c                 ó`   •— t        j                  | «      t        j                  | «      z  ‰z
  S rN   )rw   rñ  rg
  )r¡  rõ  s    €r6   Úsolve_for_kappaz=vonmises_gen.fit.<locals>.find_kappa.<locals>.solve_for_kappaô+  s#   ø€ ÜŸ6™6 %›=¬¯©°«Ñ6¸Ñ:Ð:r8   rU   r<  )r1   rE  )	rP   r¥  r!  r¤  r+   rH  r#  rR  r$  )rF   r.   rü  Úlower_boundÚupper_boundÚroot_resrõ  s         @r6   Ú
find_kappaz$vonmises_gen.fit.<locals>.find_kappaá+  sÂ   ø€ ô —‘”r—v‘v˜c D™jÓ)Ó*¬3¨t«9Ñ4ˆAð �AŠvð Ø�Q’ô;ð    1¡™g q¨¡s™m�Ø ™m�ñ # ;Ó/°1Ò4Ø&Ð&Ù$ [Ó1°QÒ6Ø&Ð&ä*¨?À8Ø4?ÀÐ3Mô O�Hà#Ÿ=™=Ð(ô —x‘x¤“×+Ñ+Ð+r8   r   )r3   rA   rC   rG  r‹   rP   rñ   rè  )rE   rF   rG   r5   rO  rö   r÷   rù  r   r.   r·  r–  s              €r6   rC   zvonmises_gen.fitË+  só   ø€ ð �8‰8�J Ô&Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ä%@ÀÀtØAEÀtó&MÑ"ˆˆf�d˜Fà�6‰6”b—e‘e�VÒä‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3ô �v‰v�d˜A¤§¡™IÓ&ˆò	(ò6	,ðr Ð&‰d©G°D«Mˆà Ð,‘±*¸TÀ3Ó2Gˆä�f‰f�Sœ2Ÿ5™5‘[ !¤b§e¡e¡)Ó,¬r¯u©uÑ4ˆØ�c˜1ˆ}Ðr8   r  )Nr‡   r   r   NNF)rƒ   r„   r…   r†   rk   rc   rÙ   r   r   rÙ  rr   rÞ   ru   rï  rò   r	   r  rK   rC   rÐ  rÑ  s   @r6   râ  râ  0+  s¤   ø„ ñ^ò~Hòó<ñ ˜MÓ*ó4ó +ð4òCòMò.ò ò
;ñ ˜}ð 5ô ð FJØ ô
Bó	ð
Bð Ù˜}ð 5/ô 0ó
Nó0ó ôNr8   râ  ræ  Úvonmises_linec                   ór   — e Zd ZdZej
                  Zd„ Zdd„Zd„ Z	d„ Z
d„ Zd„ Zd	„ Zd
„ Zd„ Zd„ Zd„ Zd„ Zy)r¬  aX  A Wald continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `wald` is:

    .. math::

        f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp(- \frac{ (x-1)^2 }{ 2x })

    for :math:`x >= 0`.

    `wald` is a special case of `invgauss` with ``mu=1``.

    %(after_notes)s

    %(example)s
    c                 ó   — g S rN   r‡   rj   s    r6   rk   zwald_gen._shape_info=,  r¦   r8   Nc                 ó*   — |j                  dd|¬«      S r“  r”  rÖ   s      r6   rÙ   zwald_gen._rvs@,  s   € Ø× Ñ   c°Ð Ó5Ð5r8   c                 ó.   — t         j                  |d«      S r  )r«  rr   r©   s     r6   rr   zwald_gen._pdfC,  s   € ä�}‰}˜Q Ó$Ð$r8   c                 ó.   — t         j                  |d«      S r  )r«  ru   r©   s     r6   ru   zwald_gen._cdfG,  ó   € Ü�}‰}˜Q Ó$Ð$r8   c                 ó.   — t         j                  |d«      S r  )r«  ry   r©   s     r6   ry   zwald_gen._sfJ,  s   € Ü�|‰|˜A˜sÓ#Ð#r8   c                 ó.   — t         j                  |d«      S r  )r«  r~   r©   s     r6   r~   zwald_gen._ppfM,  r  r8   c                 ó.   — t         j                  |d«      S r  )r«  r�   r©   s     r6   r�   zwald_gen._isfP,  r  r8   c                 ó.   — t         j                  |d«      S r  )r«  rÞ   r©   s     r6   rÞ   zwald_gen._logpdfS,  ó   € Ü×Ñ  3Ó'Ð'r8   c                 ó.   — t         j                  |d«      S r  )r«  rã   r©   s     r6   rã   zwald_gen._logcdfV,  r  r8   c                 ó.   — t         j                  |d«      S r  )r«  rç   r©   s     r6   rç   zwald_gen._logsfY,  s   € Ü�‰˜q #Ó&Ð&r8   c                  ó   — y)N)r‰   r‰   rD  rŽ  r‡   rj   s    r6   r   zwald_gen._stats\,  s   € Ø"r8   c                 ó,   — t         j                  d«      S r  )r«  rò   rj   s    r6   rò   zwald_gen._entropy_,  s   € Ü× Ñ  Ó%Ð%r8   r  )rƒ   r„   r…   r†   r   r  r  rk   rÙ   rr   ru   ry   r~   r�   rÞ   rã   rç   r   rò   r‡   r8   r6   r¬  r¬  &,  sP   „ ñð( "×4Ñ4€Mòó6ò%ò%ò$ò%ò%ò(ò(ò'ò#ó&r8   r¬  r•  c                   ó:   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
y	)
Úwrapcauchy_gena  A wrapped Cauchy continuous random variable.

    %(before_notes)s

    Notes
    -----
    The probability density function for `wrapcauchy` is:

    .. math::

        f(x, c) = \frac{1-c^2}{2\pi (1+c^2 - 2c \cos(x))}

    for :math:`0 \le x \le 2\pi`, :math:`0 < c < 1`.

    `wrapcauchy` takes ``c`` as a shape parameter for :math:`c`.

    %(after_notes)s

    %(example)s

    c                 ó   — |dkD  |dk  z  S r”  r‡   r~  s     r6   rc   zwrapcauchy_gen._argcheck|,  rK
  r8   c                 ó    — t        dddd«      gS )Nr  F)r   r   r
  r°
  rj   s    r6   rk   zwrapcauchy_gen._shape_info,  s   € Ü˜3  v¨~Ó>Ð?Ð?r8   c                 ó„   — d||z  z
  dt         j                  z  d||z  z   d|z  t        j                  |«      z  z
  z  z  S r   rú  r
  s      r6   rr   zwrapcauchy_gen._pdf‚,  s?   € à�A�a‘C‘˜!œBŸE™E™' 1 Q q¡S¡5¨¨1©¬R¯V©V°A«Y©Ñ#6Ñ7Ñ8Ð8r8   c                 óh   — d„ }d„ }d|z   d|z
  z  }t        |t        j                  k  ||f||¬«      S )Nc                 ó†   — dt         j                  z  t        j                  |t        j                  | dz  «      z  «      z  S r1  ©rP   rñ   rð  r  ©rq   Úcrs     r6   rœ  zwrapcauchy_gen._cdf.<locals>.f1ˆ,  s.   € à”R—U‘U‘7œRŸY™Y r¬"¯&©&°°1±«+¡~Ó6Ñ6Ð6r8   c           	      ó´   — ddt         j                  z  t        j                  |t        j                  dt         j                  z  | z
  dz  «      z  «      z  z
  S r1  r  r  s     r6   rë  zwrapcauchy_gen._cdf.<locals>.f2Œ,  sA   € à�qœŸ™‘w¤§¡¨2¬b¯f©f°a¼¿¹±gÀ±kÀ1±_Ó.EÑ+EÓ!FÑFÑFÐFr8   r   rŒ  )r   rP   rñ   )rE   rq   r  rœ  rë  r  s         r6   ru   zwrapcauchy_gen._cdf†,  s=   € ò	7ò	Gð �!‰e�a˜!‘e‰_ˆÜ˜!œbŸe™e™) a¨ W°°rÔ:Ð:r8   c           
      óv  — d|z
  d|z   z  }dt        j                  |t        j                  t         j                  |z  «      z  «      z  }dt         j                  z  dt        j                  |t        j                  t         j                  d|z
  z  «      z  «      z  z
  }t        j                  |dk  ||«      S )Nr‰   rU   r   r”   )rP   rð  r  rñ   rO  )rE   r}   r  rx  ÚrcqÚrcmqs         r6   r~   zwrapcauchy_gen._ppf“,  sŒ   € Ø�1‰u�s˜1‘u‰oˆØ”—	‘	˜#œbŸf™f¤R§U¡U¨1¡W›oÑ-Ó.Ñ.ˆØ”—‘‰w�qœŸ™ 3¤r§v¡v¬b¯e©e°Q°q±S©kÓ':Ñ#:Ó;Ñ;Ñ;ˆÜ�x‰x˜˜E™	 3¨Ó-Ð-r8   c                 ó`   — t        j                  dt         j                  z  d||z  z
  z  «      S r‹  rï   r~  s     r6   rò   zwrapcauchy_gen._entropy™,  s%   € Ü�v‰v�aœŸ™‘g˜q  1¡™u‘oÓ&Ð&r8   c                 óÀ   — t        |t        «      r|j                  «       }dt        j                  |«      t        j
                  |«      dt        j                  z  z  fS r»  )r?   r*   r“  rP   r‡  rý	  rñ   )rE   rF   s     r6   r•  zwrapcauchy_gen._fitstartœ,  sD   € ô �dœLÔ)Ø—>‘>Ó#ˆDØ”B—F‘F˜4“L¤"§&¡&¨£,°´"·%±%±Ñ"8Ð8Ð8r8   N)rƒ   r„   r…   r†   rc   rk   rr   ru   r~   rò   r•  r‡   r8   r6   r  r  f,  s+   „ ñò*!ò@ò9ò;ò.ò'ó9r8   r  Ú
wrapcauchyc                   óN   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zdd„Zy)Úgennorm_gena0  A generalized normal continuous random variable.

    %(before_notes)s

    See Also
    --------
    laplace : Laplace distribution
    norm : normal distribution

    Notes
    -----
    The probability density function for `gennorm` is [1]_:

    .. math::

        f(x, \beta) = \frac{\beta}{2 \Gamma(1/\beta)} \exp(-|x|^\beta),

    where :math:`x` is a real number, :math:`\beta > 0` and
    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).

    `gennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
    For :math:`\beta = 1`, it is identical to a Laplace distribution.
    For :math:`\beta = 2`, it is identical to a normal distribution
    (with ``scale=1/sqrt(2)``).

    References
    ----------

    .. [1] "Generalized normal distribution, Version 1",
           https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1

    .. [2] Nardon, Martina, and Paolo Pianca. "Simulation techniques for
           generalized Gaussian densities." Journal of Statistical
           Computation and Simulation 79.11 (2009): 1317-1329

    .. [3] Wicklin, Rick. "Simulate data from a generalized Gaussian
           distribution" in The DO Loop blog, September 21, 2016,
           https://blogs.sas.com/content/iml/2016/09/21/simulate-generalized-gaussian-sas.html

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS ©Nrn  Fr   r
  rh   rj   s    r6   rk   zgennorm_gen._shape_infoÓ,  ó   € Ü˜6 5¨1¬b¯f©f¨+°~ÓFÐGÐGr8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  ©rE   rq   rn  s      r6   rr   zgennorm_gen._pdfÖ,  s   € Ü�v‰v�d—l‘l 1 dÓ+Ó,Ð,r8   c                 ó‚   — t        j                  d|z  «      t        j                  d|z  «      z
  t	        |«      |z  z
  S r“   )rP   rð   rw   rÆ  rŽ  r(  s      r6   rÞ   zgennorm_gen._logpdfÙ,  s4   € Ü�v‰v�c˜$‘hÓ¤"§*¡*¨S°©XÓ"6Ñ6¼¸Q»À¹ÑEÐEr8   c                 óŽ   — dt        j                  |«      z  }d|z   |t        j                  d|z  t	        |«      |z  «      z  z
  S r“   )rP   rQ   rw   rÀ  rŽ  ©rE   rq   rn  r  s       r6   ru   zgennorm_gen._cdfÜ,  s?   € Ø”"—'‘'˜!“*Ñˆà�a‘˜1œrŸ|™|¨C°©H´c¸!³f¸d±lÓCÑCÑCÐCr8   c                 óŽ   — t        j                  |dz
  «      }|t        j                  d|z  d|z   d|z  |z  z
  «      d|z  z  z  S )Nr”   r‰   r¶   )rP   rQ   rw   rÈ  r+  s       r6   r~   zgennorm_gen._ppfá,  sH   € Ü�G‰G�A˜‘GÓˆà”2—?‘? 3 t¡8¨c°A©g¸¸Q¹¸q¹Ñ-@ÓAÀCÈÁHÑMÑMÐMr8   c                 ó(   — | j                  | |«      S rN   r–  r(  s      r6   ry   zgennorm_gen._sfæ,  s   € Ø�y‰y˜!˜˜TÓ"Ð"r8   c                 ó(   — | j                  ||«       S rN   rŒ  r(  s      r6   r�   zgennorm_gen._isfé,  s   € Ø—	‘	˜!˜TÓ"Ð"Ð"r8   c                 óÂ   — t        j                  d|z  d|z  d|z  g«      \  }}}dt        j                  ||z
  «      dt        j                  ||z   d|z  z
  «      dz
  fS )Nr‰   rD  r	  rˆ   r¶   )rw   rÆ  rP   r·   )rE   rn  Úc1Úc3Úc5s        r6   r   zgennorm_gen._statsì,  s_   € Ü—Z‘Z  T¡¨3¨t©8°S¸±XÐ >Ó?‰
ˆˆB�Ø”2—6‘6˜"˜r™'“? B¬¯©¨r°B©w¸¸R¹Ñ/?Ó(@À2Ñ(EÐEÐEr8   c                 óp   — d|z  t        j                  d|z  «      z
  t        j                  d|z  «      z   S r“  r|  ©rE   rn  s     r6   rò   zgennorm_gen._entropyð,  s0   € Ø�D‰yœ2Ÿ6™6 " t¡)Ó,Ñ,¬r¯z©z¸"¸t¹)Ó/DÑDÐDr8   Nc                 ó¼   — |j                  d|z  |¬«      }|d|z  z  }t        j                  |«      }|j                  |j                  ¬«      dk  }||    ||<   |S )Nr   r  r”   )rØ  rP   rû   Úrandomr·  )rE   rn  r×   rØ   r*  r�  r¢	  s          r6   rÙ   zgennorm_gen._rvsó,  sg   € ð ×Ñ˜q ™v¨DÐÓ1ˆØ�!�D‘&‰Mˆä�J‰J�q‹MˆØ×"Ñ"¨¯©Ð"Ó0°3Ñ6ˆØ�T‘7�(ˆˆ$‰Øˆr8   r  )rƒ   r„   r…   r†   rk   rr   rÞ   ru   r~   ry   r�   r   rò   rÙ   r‡   r8   r6   r#  r#  ¨,  s@   „ ñ)òTHò-òFòDò
Nò
#ò#òFòEô	r8   r#  Úgennormc                   ó@   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zy
)Úhalfgennorm_genaµ  The upper half of a generalized normal continuous random variable.

    %(before_notes)s

    See Also
    --------
    gennorm : generalized normal distribution
    expon : exponential distribution
    halfnorm : half normal distribution

    Notes
    -----
    The probability density function for `halfgennorm` is:

    .. math::

        f(x, \beta) = \frac{\beta}{\Gamma(1/\beta)} \exp(-|x|^\beta)

    for :math:`x, \beta > 0`. :math:`\Gamma` is the gamma function
    (`scipy.special.gamma`).

    `halfgennorm` takes ``beta`` as a shape parameter for :math:`\beta`.
    For :math:`\beta = 1`, it is identical to an exponential distribution.
    For :math:`\beta = 2`, it is identical to a half normal distribution
    (with ``scale=1/sqrt(2)``).

    References
    ----------

    .. [1] "Generalized normal distribution, Version 1",
           https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1

    %(example)s

    c                 ó@   — t        dddt        j                  fd«      gS r%  rh   rj   s    r6   rk   zhalfgennorm_gen._shape_info&-  r&  r8   c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  r(  s      r6   rr   zhalfgennorm_gen._pdf)-  s   € ô �v‰v�d—l‘l 1 dÓ+Ó,Ð,r8   c                 ój   — t        j                  |«      t        j                  d|z  «      z
  ||z  z
  S r  r|  r(  s      r6   rÞ   zhalfgennorm_gen._logpdf/-  s+   € Ü�v‰v�d‹|œbŸj™j¨¨T©Ó2Ñ2°Q¸±WÑ<Ð<r8   c                 ó:   — t        j                  d|z  ||z  «      S r  r¼  r(  s      r6   ru   zhalfgennorm_gen._cdf2-  s   € Ü�{‰{˜3˜t™8 Q¨¡WÓ-Ð-r8   c                 ó@   — t        j                  d|z  |«      d|z  z  S r  rë  r(  s      r6   r~   zhalfgennorm_gen._ppf5-  s    € Ü�~‰~˜c $™h¨Ó*¨S°©XÑ6Ð6r8   c                 ó:   — t        j                  d|z  ||z  «      S r  r¿  r(  s      r6   ry   zhalfgennorm_gen._sf8-  s   € Ü�|‰|˜C ™H a¨¡gÓ.Ð.r8   c                 ó@   — t        j                  d|z  |«      d|z  z  S r  r,  r(  s      r6   r�   zhalfgennorm_gen._isf;-  s    € Ü�‰˜s 4™x¨Ó+¨c°$©hÑ7Ð7r8   c                 ój   — d|z  t        j                  |«      z
  t        j                  d|z  «      z   S r  r|  r4  s     r6   rò   zhalfgennorm_gen._entropy>-  s+   € Ø�4‰xœ"Ÿ&™& ›,Ñ&¬¯©°C¸±HÓ)=Ñ=Ð=r8   NrÒ  r‡   r8   r6   r9  r9  -  s1   „ ñ"òFHò-ò=ò.ò7ò/ò8ó>r8   r9  Úhalfgennormc                   óR   ‡ — e Zd ZdZd„ Zd„ Zˆ fd„Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zd
„ Zˆ xZS )Úcrystalball_gena¹  
    Crystalball distribution

    %(before_notes)s

    Notes
    -----
    The probability density function for `crystalball` is:

    .. math::

        f(x, \beta, m) =  \begin{cases}
                            N \exp(-x^2 / 2),  &\text{for } x > -\beta\\
                            N A (B - x)^{-m}  &\text{for } x \le -\beta
                          \end{cases}

    where :math:`A = (m / |\beta|)^m  \exp(-\beta^2 / 2)`,
    :math:`B = m/|\beta| - |\beta|` and :math:`N` is a normalisation constant.

    `crystalball` takes :math:`\beta > 0` and :math:`m > 1` as shape
    parameters.  :math:`\beta` defines the point where the pdf changes
    from a power-law to a Gaussian distribution.  :math:`m` is the power
    of the power-law tail.

    %(after_notes)s

    .. versionadded:: 0.19.0

    References
    ----------
    .. [1] "Crystal Ball Function",
           https://en.wikipedia.org/wiki/Crystal_Ball_function

    %(example)s
    c                 ó   — |dkD  |dkD  z  S )z@
        Shape parameter bounds are m > 1 and beta > 0.
        r   r   r‡   )rE   rn  r?  s      r6   rc   zcrystalball_gen._argchecki-  s   € ð �A‘˜$ ™(Ñ#Ð#r8   c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS )Nrn  Fr   r
  r?  r   rh   )rE   ÚibetaÚims      r6   rk   zcrystalball_gen._shape_infoo-  s<   € Ü˜6 5¨1¬b¯f©f¨+°~ÓFˆÜ˜˜U Q¬¯© K°Ó@ˆØ�rˆ{Ðr8   c                 ó&   •— t         ‰| �  |d¬«      S )N)r   rÊ  rM  rS  rT  s     €r6   r•  zcrystalball_gen._fitstartt-  s   ø€ ä‰wÑ  ¨HÐ Ó5Ð5r8   c                 ó¼   — d||z  |dz
  z  t        j                  |dz   dz  «      z  t        t        |«      z  z   z  }d„ }d„ }|t	        || kD  |||f||¬«      z  S )a`  
        Return PDF of the crystalball function.

                                            --
                                           | exp(-x**2 / 2),  for x > -beta
        crystalball.pdf(x, beta, m) =  N * |
                                           | A * (B - x)**(-m), for x <= -beta
                                            --
        r‰   r   rU   r¶   c                 ó:   — t        j                  | dz   dz  «      S r  ru  ©rq   rn  r?  s      r6   Úrhsz!crystalball_gen._pdf.<locals>.rhs…-  s   € Ü—6‘6˜1˜a™4˜% !™)Ó$Ð$r8   c                 ól   — ||z  |z  t        j                  |dz   dz  «      z  ||z  |z
  | z
  | z  z  S rµ   ru  rL  s      r6   Úlhsz!crystalball_gen._pdf.<locals>.lhsˆ-  sF   € Ø�t‘V˜a‘K¤"§&¡&¨$°©'¨°C©Ó"8Ñ8Ø�t‘V˜d‘] QÑ&¨1¨"Ñ-ñ.ð /r8   rŒ  ©rP   r·   r¸   rÀ   r   ©rE   rq   rn  r?  rë  rM  rO  s          r6   rr   zcrystalball_gen._pdfx-  ss   € ð �1�T‘6˜Q˜q™S‘>¤B§F¡F¨D°!©G¨8°c©>Ó$:Ñ:Ü¤¨4£Ñ0ñ1ñ 2ˆò	%ò	/ð ”:˜a 4 %™i¨!¨T°1¨¸ÀÔEÑEÐEr8   c                 óâ   — d||z  |dz
  z  t        j                  |dz   dz  «      z  t        t        |«      z  z   z  }d„ }d„ }t        j                  |«      t        || kD  |||f||¬«      z   S )zH
        Return the log of the PDF of the crystalball function.
        r‰   r   rU   r¶   c                 ó   — | dz   dz  S r  r‡   rL  s      r6   rM  z$crystalball_gen._logpdf.<locals>.rhs•-  s   € Ø�q‘D�5˜‘7ˆNr8   c                 óŽ   — |t        j                  ||z  «      z  |dz  dz  z
  |t        j                  ||z  |z
  | z
  «      z  z
  S r  r2  rL  s      r6   rO  z$crystalball_gen._logpdf.<locals>.lhs˜-  sF   € Ø”R—V‘V˜A˜d™F“^Ñ# d¨A¡g¨a¡iÑ/°!´B·F±F¸1¸T¹6ÀD¹=È1Ñ;LÓ4MÑ2MÑMÐMr8   rŒ  )rP   r·   r¸   rÀ   rð   r   rQ  s          r6   rÞ   zcrystalball_gen._logpdfŽ-  s|   € ð �1�T‘6˜Q˜q™S‘>¤B§F¡F¨D°!©G¨8°c©>Ó$:Ñ:Ü¤¨4£Ñ0ñ1ñ 2ˆò	ò	Nô �v‰v�a‹yœ: a¨4¨%¡i°!°T¸1°ÀÈÔMÑMÐMr8   c                 ó¼   — d||z  |dz
  z  t        j                  |dz   dz  «      z  t        t        |«      z  z   z  }d„ }d„ }|t	        || kD  |||f||¬«      z  S )z8
        Return CDF of the crystalball function
        r‰   r   rU   r¶   c                 ó’   — ||z  t        j                  |dz   dz  «      z  |dz
  z  t        t        | «      t        | «      z
  z  z   S ©NrU   r¶   r   ©rP   r·   r¸   rÀ   rL  s      r6   rM  z!crystalball_gen._cdf.<locals>.rhs¤-  sN   € Ø�t‘VœrŸv™v t¨Q¡w h°¡nÓ5Ñ5¸¸1¹Ñ=Ü¤9¨Q£<´)¸T¸EÓ2BÑ#BÑCñDð Er8   c                 ó~   — ||z  |z  t        j                  |dz   dz  «      z  ||z  |z
  | z
  | dz   z  z  |dz
  z  S rW  ru  rL  s      r6   rO  z!crystalball_gen._cdf.<locals>.lhs¨-  sV   € Ø�t‘V˜a‘K¤"§&¡&¨$°©'¨°C©Ó"8Ñ8Ø�t‘V˜d‘] QÑ&¨1¨"¨Q©$Ñ/ñ0Ø34°Q±3ñ8ð 9r8   rŒ  rP  rQ  s          r6   ru   zcrystalball_gen._cdf�-  st   € ð �1�T‘6˜Q˜q™S‘>¤B§F¡F¨D°!©G¨8°c©>Ó$:Ñ:Ü¤¨4£Ñ0ñ1ñ 2ˆò	Eò	9ð ”:˜a 4 %™i¨!¨T°1¨¸ÀÔEÑEÐEr8   c                 ó@   ‡ — d„ }ˆ fd„}t        || kD  |||f||¬«      S )zD
        Survival function of the crystalball distribution.
        c                 ó¢   — ||z  |dz
  z  t        j                  |dz   dz  «      z  t        t        |«      z  z   }t        t	        | «      z  |z  S r1  )rP   r·   r¸   rÀ   rÊ   )rq   rn  r?  ÚMs       r6   rM  z crystalball_gen._sf.<locals>.rhs³-  sM   € à�$‘˜˜A™‘œrŸv™v t¨Q¡w h¨q¡jÓ1Ñ1´KÄ	È$ÃÑ4OÑOˆAÜœx¨›{Ñ*¨1Ñ,Ð,r8   c                 ó0   •— d‰j                  | ||«      z
  S r^   r–  )rq   rn  r?  rE   s      €r6   rO  z crystalball_gen._sf.<locals>.lhs¸-  s   ø€ à�t—y‘y  D¨!Ó,Ñ,Ð,r8   rŒ  rì  )rE   rq   rn  r?  rM  rO  s   `     r6   ry   zcrystalball_gen._sf®-  s,   ø€ ò
	-ô
	-ô ˜!˜t˜e™) a¨¨q \°S¸SÔAÐAr8   c                 ó
  — d||z  |dz
  z  t        j                  |dz   dz  «      z  t        t        |«      z  z   z  }|||z  z  t        j                  |dz   dz  «      z  |dz
  z  }d„ }d„ }t	        ||k  |||f||¬«      S )Nr‰   r   rU   r¶   c                 óÚ   — t        j                  |dz   dz  «      }||z  |z  |dz
  z  }d|t        t        |«      z  z   z  }||z  |z
  |dz
  ||z  | z  z  |z  | z  |z  dd|z
  z  z  z
  S r‹  rX  ©rô  rn  r?  Úeb2rC  rë  s         r6   Úppf_lessz&crystalball_gen._ppf.<locals>.ppf_lessÃ-  s‘   € Ü—&‘&˜$ ™'˜ !™Ó$ˆCØ�4‘˜3‘ ! A¡#Ñ&ˆAØ�1”{¤Y¨t£_Ñ4Ñ4Ñ5ˆAØ�d‘F˜T‘MØ˜!‘e˜a ™f¨¨™^Ñ+¨CÑ/°Ñ1°!Ñ3°q¸!¸A¹#±wÑ?ñ@ð Ar8   c                 óÖ   — t        j                  |dz   dz  «      }||z  |z  |dz
  z  }d|t        t        |«      z  z   z  }t	        t        | «      dt        z  | |z  |z
  z  z   «      S r‹  )rP   r·   r¸   rÀ   rÇ   r`  s         r6   Úppf_greaterz)crystalball_gen._ppf.<locals>.ppf_greaterÊ-  sp   € Ü—&‘&˜$ ™'˜ !™Ó$ˆCØ�4‘˜3‘ ! A¡#Ñ&ˆAØ�1”{¤Y¨t£_Ñ4Ñ4Ñ5ˆAÜœY¨ uÓ-°´;±ÀÀ1ÁÀqÁÑ0IÑIÓJÐJr8   rŒ  rP  )rE   rô  rn  r?  rë  Úpbetarb  rd  s           r6   r~   zcrystalball_gen._ppf¾-  sš   € Ø�1�T‘6˜Q˜q™S‘>¤B§F¡F¨D°!©G¨8°c©>Ó$:Ñ:Ü¤¨4£Ñ0ñ1ñ 2ˆà�Q�t‘V‘œrŸv™v t¨Q¡w h¨q¡jÓ1Ñ1°Q¸±UÑ;ˆò	Aò	Kô ˜!˜e™) a¨¨q \°XÀ+ÔNÐNr8   c           	      ó  — d||z  |dz
  z  t        j                  |dz   dz  «      z  t        t        |«      z  z   z  }d„ }|t	        |dz   |k  |||ft        j
                  |t         j                  g¬«      t         j                  «      z  S )zR
        Returns the n-th non-central moment of the crystalball function.
        r‰   r   rU   r¶   c                 ó
  — ||z  |z  t        j                  |dz   dz  «      z  }||z  |z
  }d| dz
  dz  z  t        j                  | dz   dz  «      z  dd| z  t        j                  | dz   dz  |dz  dz  «      z  z   z  }t        j
                  |j                  «      }t        t        | «      dz   «      D ]C  }|t        j                  | |«      || |z
  z  z  d|z  z  ||z
  dz
  z  ||z  | |z   dz   z  z  z  }ŒE ||z  |z   S )zƒ
            Returns n-th moment. Defined only if n+1 < m
            Function cannot broadcast due to the loop over n
            rU   r¶   r   r‰   r¿  )
rP   r·   rw   rØ  r½  rä  r·  rý  r  Úbinom)rb   rn  r?  rm  rn  rM  rO  r  s           r6   r{  z*crystalball_gen._munp.<locals>.n_th_momentÙ-  s!  € ð
 �4‘˜!‘œbŸf™f d¨A¡g X°¡^Ó4Ñ4ˆAØ�$‘˜‘ˆAØ˜˜!™˜S‘y‘>¤B§H¡H¨a°©c°1©WÓ$5Ñ5Ø˜2 ™'¤B§K¡K°°1±°a±¸¸q¹À1¹Ó$EÑEÑEñGˆCä—(‘(˜3Ÿ9™9Ó%ˆCÜœ3˜q›6 A™:Ó&ò 0�ØœŸ™  A›¨¨Q¨q©S©Ñ1°R¸!±GÑ;¸qÀ1¹uÀq¹yÑIØ˜4™ A 2¨¡6¨A¡:Ñ.ñ/ñ 0‘ð0ð �s‘7˜S‘=Ð r8   r“  )rP   r·   r¸   rÀ   r   r�  r–  ri   )rE   rb   rn  r?  rë  r{  s         r6   r  zcrystalball_gen._munpÒ-  s�   € ð �1�T‘6˜Q˜q™S‘>¤B§F¡F¨D°!©G¨8°c©>Ó$:Ñ:Ü¤¨4£Ñ0ñ1ñ 2ˆò	!ð ”:˜a !™e a™i¨!¨T°1¨Ü Ÿl™l¨;ÄÇ
Á
¸|ÔLÜ Ÿf™fó&ñ &ð 	&r8   )rƒ   r„   r…   r†   rc   rk   r•  rr   rÞ   ru   ry   r~   r  rÐ  rÑ  s   @r6   rD  rD  E-  s;   ø„ ñ"òF$òô
6òFò,NòFò"Bò Oö(&r8   rD  ÚcrystalballzA Crystalball Function)r�   Úlongnamec                 ó@   — t        j                  d| dz  dz  «      dz  S )aµ  
    Utility function for the argus distribution used in the pdf, sf and
    moment calculation.
    Note that for all x > 0:
    gammainc(1.5, x**2/2) = 2 * (_norm_cdf(x) - x * _norm_pdf(x) - 0.5).
    This can be verified directly by noting that the cdf of Gamma(1.5) can
    be written as erf(sqrt(x)) - 2*sqrt(x)*exp(-x)/sqrt(Pi).
    We use gammainc instead of the usual definition because it is more precise
    for small chi.
    rÊ  rU   r¼  )rÙ  s    r6   Ú
_argus_phirl  ð-  s"   € ô �;‰;�s˜C ™F 1™HÓ%¨Ñ)Ð)r8   c                   óD   — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zdd„Z	dd	„Z
d
„ Zy)Ú	argus_gena  
    Argus distribution

    %(before_notes)s

    Notes
    -----
    The probability density function for `argus` is:

    .. math::

        f(x, \chi) = \frac{\chi^3}{\sqrt{2\pi} \Psi(\chi)} x \sqrt{1-x^2}
                     \exp(-\chi^2 (1 - x^2)/2)

    for :math:`0 < x < 1` and :math:`\chi > 0`, where

    .. math::

        \Psi(\chi) = \Phi(\chi) - \chi \phi(\chi) - 1/2

    with :math:`\Phi` and :math:`\phi` being the CDF and PDF of a standard
    normal distribution, respectively.

    `argus` takes :math:`\chi` as shape a parameter. Details about sampling
    from the ARGUS distribution can be found in [2]_.

    %(after_notes)s

    References
    ----------
    .. [1] "ARGUS distribution",
           https://en.wikipedia.org/wiki/ARGUS_distribution
    .. [2] Christoph Baumgarten "Random variate generation by fast numerical
           inversion in the varying parameter case." Research in Statistics,
           vol. 1, 2023, doi:10.1080/27684520.2023.2279060.

    .. versionadded:: 0.19.0

    %(example)s
    c                 ó@   — t        dddt        j                  fd«      gS )NrÙ  Fr   r
  rh   rj   s    r6   rk   zargus_gen._shape_info'.  ó   € Ü˜5 %¨!¬R¯V©V¨°nÓEÐFÐFr8   c                 óh  — t        j                  d¬«      5  d||z  z
  }dt        j                  |«      z  t        z
  t        j                  t	        |«      «      z
  }|t        j                  |«      z   dt        j
                  | |z  «      z  z   |dz  |z  dz  z
  cd d d «       S # 1 sw Y   y xY w)Nr9  r:  r‰   r†  r”   rU   )rP   r<  rð   r¼   rl  r¦  )rE   rq   rÙ  r�  rm  s        r6   rÞ   zargus_gen._logpdf*.  s”   € ä�[‰[ Ô)ñ 	GØ�a˜‘c‘	ˆAØ”"—&‘&˜“+‘¤Ñ.´·±¼
À3»Ó1HÑHˆAØ”r—v‘v˜a“y‘= 3¤r§x¡x°°°1±£~Ñ#5Ñ5¸¸Q¹À¹
ÀQ¹ÑF÷	G÷ 	Gò 	Gús   —BB(Â(B1c                 óL   — t        j                  | j                  ||«      «      S rN   rÝ  ©rE   rq   rÙ  s      r6   rr   zargus_gen._pdf1.  s   € Ü�v‰v�d—l‘l 1 cÓ*Ó+Ð+r8   c                 ó,   — d| j                  ||«      z
  S r  r!  rs  s      r6   ru   zargus_gen._cdf4.  s   € Ø�T—X‘X˜a Ó%Ñ%Ð%r8   c                 ón   — t        |t        j                  d|z
  d|z   z  «      z  «      t        |«      z  S r^   )rl  rP   rÿ   rs  s      r6   ry   zargus_gen._sf7.  s0   € Ü˜#¤§¡¨¨Q©°°Q±©Ó 8Ñ8Ó9¼JÀs»OÑKÐKr8   Nc                 óZ  ‡	‡
— t        j                  |«      }|j                  dk(  r| j                  |||¬«      }nåt	        |j
                  |«      \  }Š	t        t        j                  |«      «      }t        j                  |«      }t        j                  |gdgdgg¬«      Š
‰
j                  sqt        ˆ	ˆ
fd„t        t        |«       d«      D «       «      }| j                  ‰
d   ||¬«      }|j                  |«      ||<   ‰
j                  «        ‰
j                  sŒq|dk(  r|d   }|S )	Nr   )rÙ  rØ   rÅ  rÆ  rÇ  c              3   ó\   •K  — | ]#  }‰|   s‰j                   |   n
t        d «      –— Œ% y ­wrN   rË  rÍ  s     €€r6   rµ  z!argus_gen._rvs.<locals>.<genexpr>G.  rÐ  rÑ  r   r‡   )rP   rû   r×   rÒ  r   r·  r  rü  rÓ  rÔ  rÕ  rÖ  rý  r¤  rN  r×  )rE   rÙ  r×   rØ   rö  rØ  rÙ  rÚ  rõ  rÎ  rÏ  s            @@r6   rÙ   zargus_gen._rvs:.  s  ù€ Ü�j‰j˜‹oˆØ�8‰8�qŠ=Ø×"Ñ" 3°4Ø0<ð #ó >‰Cô # 3§9¡9¨dÓ3‰GˆC�ÜœRŸW™W S›\Ó*ˆJÜ—(‘(˜4“.ˆCÜ—‘˜C˜5Ø"/ Ø&0 \ Nô4ˆBð —k’kÜô ;Ü%*¬C°«I¨:°qÓ%9ô;ó ;�à×$Ñ$ R¨¡U°zØ2>ð %ó @�àŸ9™9 S›>��C‘Ø—‘”ð —k“kð �2Š:Ø�b‘'ˆCØˆ
r8   c                 ó–  — t        t        j                  |«      «      }t        t        j                  |«      «      }t        j
                  |«      }d}||z  }|dk  rŸ| dz  }	||k  �rË||z
  }
|j                  |
¬«      }|j                  |
¬«      }|dz  }t        j                  |«      |	|z  k  }t        j                  |«      }|dkD  r(t        j                  d||   z
  «      }|||||z    ||z  }||k  rŒ‘�n8|dk  rÀt        j                  | dz  «      }||k  �r||z
  }
|j                  |
¬«      }|j                  |
¬«      }dt        j                  |d|z
  z  |z   «      z  |z  }|dz  |z   dk  }t        j                  |«      }|dkD  r(t        j                  d||   z   «      }|||||z    ||z  }||k  rŒ ns||k  rP||z
  }
|j                  d|
¬«      }||dz  k  }t        j                  |«      }|dkD  r||   ||||z    ||z  }||k  rŒPt        j                  dd|z  |z  z
  «      }t        j                  ||«      S )	Nr   r”   rU   r  rŠ  r   gÍÌÌÌÌÌü?rÊ  )rÖ  rP   rã  r  rü  rä  r  rð   r¥  rÿ   r·   r%  rN  )rE   rÙ  rÙ  rØ   rê  rë  rq   rì  r´  rÎ  r  r  rË  r*  rú  rû  rÙ  Úechirõ  s                      r6   rÒ  zargus_gen._rvs_scalarR.  sq  € ôh ”r—}‘} ZÓ0Ó1ˆÜ”—‘˜“Ó ˆÜ�H‰H�Q‹KˆØˆ	Ø�S‰yˆØ�#Š:Ø�˜‘	ˆAØ˜a“-Ø˜	‘M�Ø ×(Ñ(¨aÐ(Ó0�Ø ×(Ñ(¨aÐ(Ó0�Ø˜‘H�äŸ&™& ›) q¨1¡uÑ,�ÜŸV™V F›^�
Ø ’>äŸ'™' ! a¨¡i¡-Ó0�CØ<?�A�i ¨ZÑ!7Ð9Ø Ñ+�Ið ˜a•-ð �CŠZÜ—6‘6˜4˜% !™)Ó$ˆDØ˜a“-Ø˜	‘M�Ø ×(Ñ(¨aÐ(Ó0�Ø ×(Ñ(¨aÐ(Ó0�ØœŸ™˜t q¨1¡u™~°Ñ1Ó2Ñ2°TÑ9�ð ˜Q™$ ™( a™-�ÜŸV™V F›^�
Ø ’>ÜŸ'™' ! a¨¡i¡-Ó0�CØ<?�A�i ¨ZÑ!7Ð9Ø Ñ+�Ið ˜a”-ð ˜a’-Ø˜	‘M�Ø ×/Ñ/°¸!Ð/Ó<�Ø˜t a™x™-�ÜŸV™V F›^�
Ø ’>Ø<=¸f¹I�A�i ¨ZÑ!7Ð9Ø Ñ+�Ið ˜a“-ô —‘˜˜A ™E D™LÑ(Ó)ˆAä�z‰z˜!˜VÓ$Ð$r8   c                 óÄ  — t        j                  |t        ¬«      }t        |«      }t        j                  t         j
                  dz  «      |z  t        j                  d|dz  dz  «      z  |z  }t        j                  |«      }|dkD  }||   }dd|dz  z  z
  |t        |«      z  ||   z  z   ||<   ||    }g d¢}t        j                  ||«      || <   |||dz  z
  d d fS )	Nr  r.  r   rU   r$  gš™™™™™¹?r†  )	g„_1gªÛÖ¾r   gWB³éa¿r   g½p|R÷H?r   gE'«å�¡?r   gš™™™™™Ù?)rP   rû   rR  rl  rÿ   rñ   rw   r	  rl  rº   rI  )rE   rÙ  rñ  r?  rD  r¢	  r  Úcoefs           r6   r   zargus_gen._stats·.  sÞ   € ô �j‰j˜¤EÔ*ˆÜ˜‹oˆÜ�G‰G”B—E‘E˜!‘GÓ˜sÑ"¤R§V¡V¨A¨s°A©v°a©xÓ%8Ñ8¸3Ñ>ˆä�m‰m˜CÓ ˆØ�S‰yˆØ�‰IˆØ˜˜A˜q™D™‘L 1¤y°£|Ñ#3°c¸$±iÑ#?Ñ?ˆˆD‰	Ø��‰JˆÚKˆÜ—Z‘Z  aÓ(ˆˆTˆE‰
Ø�#˜˜1™‘*˜d DÐ(Ð(r8   r  )rƒ   r„   r…   r†   rk   rÞ   rr   ru   ry   rÙ   rÒ  r   r‡   r8   r6   rn  rn  þ-  s5   „ ñ'òPGòGò,ò&òLóó0c%óJ)r8   rn  ÚarguszAn Argus Function)r�   rj  r‹   rŒ   c                   óh   ‡ — e Zd ZdZej
                  Zddœˆ fd„
Zd„ Zd„ Zd„ Z	d„ Z
d	„ Zˆ fd
„Zˆ xZS )Úrv_histograma3  
    Generates a distribution given by a histogram.
    This is useful to generate a template distribution from a binned
    datasample.

    As a subclass of the `rv_continuous` class, `rv_histogram` inherits from it
    a collection of generic methods (see `rv_continuous` for the full list),
    and implements them based on the properties of the provided binned
    datasample.

    Parameters
    ----------
    histogram : tuple of array_like
        Tuple containing two array_like objects.
        The first containing the content of n bins,
        the second containing the (n+1) bin boundaries.
        In particular, the return value of `numpy.histogram` is accepted.

    density : bool, optional
        If False, assumes the histogram is proportional to counts per bin;
        otherwise, assumes it is proportional to a density.
        For constant bin widths, these are equivalent, but the distinction
        is important when bin widths vary (see Notes).
        If None (default), sets ``density=True`` for backwards compatibility,
        but warns if the bin widths are variable. Set `density` explicitly
        to silence the warning.

        .. versionadded:: 1.10.0

    Notes
    -----
    When a histogram has unequal bin widths, there is a distinction between
    histograms that are proportional to counts per bin and histograms that are
    proportional to probability density over a bin. If `numpy.histogram` is
    called with its default ``density=False``, the resulting histogram is the
    number of counts per bin, so ``density=False`` should be passed to
    `rv_histogram`. If `numpy.histogram` is called with ``density=True``, the
    resulting histogram is in terms of probability density, so ``density=True``
    should be passed to `rv_histogram`. To avoid warnings, always pass
    ``density`` explicitly when the input histogram has unequal bin widths.

    There are no additional shape parameters except for the loc and scale.
    The pdf is defined as a stepwise function from the provided histogram.
    The cdf is a linear interpolation of the pdf.

    .. versionadded:: 0.19.0

    Examples
    --------

    Create a scipy.stats distribution from a numpy histogram

    >>> import scipy.stats
    >>> import numpy as np
    >>> data = scipy.stats.norm.rvs(size=100000, loc=0, scale=1.5,
    ...                             random_state=123)
    >>> hist = np.histogram(data, bins=100)
    >>> hist_dist = scipy.stats.rv_histogram(hist, density=False)

    Behaves like an ordinary scipy rv_continuous distribution

    >>> hist_dist.pdf(1.0)
    0.20538577847618705
    >>> hist_dist.cdf(2.0)
    0.90818568543056499

    PDF is zero above (below) the highest (lowest) bin of the histogram,
    defined by the max (min) of the original dataset

    >>> hist_dist.pdf(np.max(data))
    0.0
    >>> hist_dist.cdf(np.max(data))
    1.0
    >>> hist_dist.pdf(np.min(data))
    7.7591907244498314e-05
    >>> hist_dist.cdf(np.min(data))
    0.0

    PDF and CDF follow the histogram

    >>> import matplotlib.pyplot as plt
    >>> X = np.linspace(-5.0, 5.0, 100)
    >>> fig, ax = plt.subplots()
    >>> ax.set_title("PDF from Template")
    >>> ax.hist(data, density=True, bins=100)
    >>> ax.plot(X, hist_dist.pdf(X), label='PDF')
    >>> ax.plot(X, hist_dist.cdf(X), label='CDF')
    >>> ax.legend()
    >>> fig.show()

    N)Údensityc                ót  •— || _         || _        t        |«      dk7  rt        d«      ‚t	        j
                  |d   «      | _        t	        j
                  |d   «      | _        t        | j                  «      dz   t        | j                  «      k7  rt        d«      ‚| j                  dd | j                  dd z
  | _        t	        j                  | j                  | j                  d   «       }|€#|r!d}t        j                  |t        d¬	«       d
}n |s| j                  | j                  z  | _        | j                  t        t	        j                  | j                  | j                  z  «      «      z  | _        t	        j                  | j                  | j                  z  «      | _        t	        j"                  d| j                  dg«      | _        t	        j"                  d| j                   g«      | _        | j                  d   x|d<   | _        | j                  d   x|d<   | _        t)        ‰| �T  |i |¤Ž y)a5  
        Create a new distribution using the given histogram

        Parameters
        ----------
        histogram : tuple of array_like
            Tuple containing two array_like objects.
            The first containing the content of n bins,
            the second containing the (n+1) bin boundaries.
            In particular, the return value of np.histogram is accepted.
        density : bool, optional
            If False, assumes the histogram is proportional to counts per bin;
            otherwise, assumes it is proportional to a density.
            For constant bin widths, these are equivalent.
            If None (default), sets ``density=True`` for backward
            compatibility, but warns if the bin widths are variable. Set
            `density` explicitly to silence the warning.
        rU   z)Expected length 2 for parameter histogramr   r   zbNumber of elements in histogram content and histogram boundaries do not match, expected n and n+1.Nr¿  zjBin widths are not constant. Assuming `density=True`.Specify `density` explicitly to silence this warning.rE  Trˆ   r‹   rŒ   )Ú
_histogramÚ_densityr¤  rú   rP   rû   Ú_hpdfÚ_hbinsÚ_hbin_widthsÚallcloserH  rI  rJ  rR  r¥  ÚcumsumÚ_hcdfÚhstackr‹   rŒ   rA   rP  )rE   Ú	histogramr  rG   r  Ú	bins_varyrL  r–  s          €r6   rP  zrv_histogram.__init__)/  sÊ  ø€ ð& $ˆŒØˆŒÜˆy‹>˜QÒÜÐHÓIÐIÜ—Z‘Z 	¨!¡Ó-ˆŒ
Ü—j‘j ¨1¡Ó.ˆŒÜˆt�z‰z‹?˜QÑ¤# d§k¡kÓ"2Ò2Üð 3ó 4ð 4ð !ŸK™K¨¨˜O¨d¯k©k¸#¸2Ð.>Ñ>ˆÔÜŸ™ D×$5Ñ$5°t×7HÑ7HÈÑ7KÓLÐLˆ	Øˆ?™yðOˆGä�M‰M˜'¤>¸aÕ@Ø‰GÙØŸ™ d×&7Ñ&7Ñ7ˆDŒJà—Z‘Z¤%¬¯©¨t¯z©z¸D×<MÑ<MÑ/MÓ(NÓ"OÑOˆŒ
Ü—Y‘Y˜tŸz™z¨D×,=Ñ,=Ñ=Ó>ˆŒ
Ü—Y‘Y  T§Z¡Z°Ð5Ó6ˆŒ
Ü—Y‘Y  T§Z¡ZÐ0Ó1ˆŒ
à#Ÿ{™{¨1™~Ð-ˆˆs‰�d”fØ#Ÿ{™{¨2™Ð.ˆˆs‰�d”fÜ‰Ñ˜$Ð) &Ó)r8   c                 ó`   — | j                   t        j                  | j                  |d¬«         S )z&
        PDF of the histogram
        r'  )Úside)rƒ  rP   Úsearchsortedr„  r©   s     r6   rr   zrv_histogram._pdfY/  s$   € ð �z‰zœ"Ÿ/™/¨$¯+©+°q¸wÔGÑHÐHr8   c                 óX   — t        j                  || j                  | j                  «      S )z3
        CDF calculated from the histogram
        )rP   Úinterpr„  rˆ  r©   s     r6   ru   zrv_histogram._cdf_/  s   € ô �y‰y˜˜DŸK™K¨¯©Ó4Ð4r8   c                 óX   — t        j                  || j                  | j                  «      S )zC
        Percentile function calculated from the histogram
        )rP   r�  rˆ  r„  r©   s     r6   r~   zrv_histogram._ppfe/  s   € ô �y‰y˜˜DŸJ™J¨¯©Ó4Ð4r8   c                 ó®   — | j                   dd |dz   z  | j                   dd |dz   z  z
  |dz   z  }t        j                  | j                  dd |z  «      S )z$Compute the n-th non-central moment.r   Nr¿  )r„  rP   r¥  rƒ  )rE   rb   Ú	integralss      r6   r  zrv_histogram._munpk/  s[   € à—[‘[  �_ q¨¡sÑ+¨d¯k©k¸#¸2Ð.>ÀÀ1ÁÑ.EÑEÈ!ÈAÉ#ÑNˆ	Ü�v‰v�d—j‘j  2Ð&¨Ñ2Ó3Ð3r8   c                 óÜ   — t        | j                  dd dkD  | j                  dd ft        j                  d«      }t        j                  | j                  dd |z  | j
                  z  «       S )zCompute entropy of distributionr   r¿  rˆ   )r   rƒ  rP   rð   r¥  r…  )rE   rõ  s     r6   rò   zrv_histogram._entropyp/  sh   € ä˜Ÿ™ A bÐ)¨CÑ/ØŸ*™* Q rÐ*Ð,ÜŸ™Øóˆô —‘�t—z‘z ! BÐ'¨#Ñ-°×0AÑ0AÑAÓBÐBÐBr8   c                 ó`   •— t         ‰| �  «       }| j                  |d<   | j                  |d<   |S )zF
        Set the histogram as additional constructor argument
        rŠ  r  )rA   Ú_updated_ctor_paramr�  r‚  )rE   Údctr–  s     €r6   r–  z rv_histogram._updated_ctor_paramx/  s2   ø€ ô ‰gÑ)Ó+ˆØŸ?™?ˆˆKÑØŸ™ˆˆI‰Øˆ
r8   )rƒ   r„   r…   r†   r   r  rP  rr   ru   r~   r  rò   r–  rÐ  rÑ  s   @r6   r~  r~  Ë.  sE   ø„ ñZðv "×/Ñ/€Mà15ö .*ò`Iò5ò5ò4ò
C÷ð r8   r~  c                   ó@   ‡ — e Zd ZdZd„ Zd„ Zˆ fd„Zd„ Zd„ Zd„ Z	ˆ xZ
S )Ústudentized_range_genuO  A studentized range continuous random variable.

    %(before_notes)s

    See Also
    --------
    t: Student's t distribution

    Notes
    -----
    The probability density function for `studentized_range` is:

    .. math::

         f(x; k, \nu) = \frac{k(k-1)\nu^{\nu/2}}{\Gamma(\nu/2)
                        2^{\nu/2-1}} \int_{0}^{\infty} \int_{-\infty}^{\infty}
                        s^{\nu} e^{-\nu s^2/2} \phi(z) \phi(sx + z)
                        [\Phi(sx + z) - \Phi(z)]^{k-2} \,dz \,ds

    for :math:`x â‰¥ 0`, :math:`k > 1`, and :math:`\nu > 0`.

    `studentized_range` takes ``k`` for :math:`k` and ``df`` for :math:`\nu`
    as shape parameters.

    When :math:`\nu` exceeds 100,000, an asymptotic approximation (infinite
    degrees of freedom) is used to compute the cumulative distribution
    function [4]_ and probability distribution function.

    %(after_notes)s

    References
    ----------

    .. [1] "Studentized range distribution",
           https://en.wikipedia.org/wiki/Studentized_range_distribution
    .. [2] Batista, Ben DÃªivide, et al. "Externally Studentized Normal Midrange
           Distribution." CiÃªncia e Agrotecnologia, vol. 41, no. 4, 2017, pp.
           378-389., doi:10.1590/1413-70542017414047716.
    .. [3] Harter, H. Leon. "Tables of Range and Studentized Range." The Annals
           of Mathematical Statistics, vol. 31, no. 4, 1960, pp. 1122-1147.
           JSTOR, www.jstor.org/stable/2237810. Accessed 18 Feb. 2021.
    .. [4] Lund, R. E., and J. R. Lund. "Algorithm AS 190: Probabilities and
           Upper Quantiles for the Studentized Range." Journal of the Royal
           Statistical Society. Series C (Applied Statistics), vol. 32, no. 2,
           1983, pp. 204-210. JSTOR, www.jstor.org/stable/2347300. Accessed 18
           Feb. 2021.

    Examples
    --------
    >>> import numpy as np
    >>> from scipy.stats import studentized_range
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)

    Display the probability density function (``pdf``):

    >>> k, df = 3, 10
    >>> x = np.linspace(studentized_range.ppf(0.01, k, df),
    ...                 studentized_range.ppf(0.99, k, df), 100)
    >>> ax.plot(x, studentized_range.pdf(x, k, df),
    ...         'r-', lw=5, alpha=0.6, label='studentized_range pdf')

    Alternatively, the distribution object can be called (as a function)
    to fix the shape, location and scale parameters. This returns a "frozen"
    RV object holding the given parameters fixed.

    Freeze the distribution and display the frozen ``pdf``:

    >>> rv = studentized_range(k, df)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

    Check accuracy of ``cdf`` and ``ppf``:

    >>> vals = studentized_range.ppf([0.001, 0.5, 0.999], k, df)
    >>> np.allclose([0.001, 0.5, 0.999], studentized_range.cdf(vals, k, df))
    True

    Rather than using (``studentized_range.rvs``) to generate random variates,
    which is very slow for this distribution, we can approximate the inverse
    CDF using an interpolator, and then perform inverse transform sampling
    with this approximate inverse CDF.

    This distribution has an infinite but thin right tail, so we focus our
    attention on the leftmost 99.9 percent.

    >>> a, b = studentized_range.ppf([0, .999], k, df)
    >>> a, b
    0, 7.41058083802274

    >>> from scipy.interpolate import interp1d
    >>> rng = np.random.default_rng()
    >>> xs = np.linspace(a, b, 50)
    >>> cdf = studentized_range.cdf(xs, k, df)
    # Create an interpolant of the inverse CDF
    >>> ppf = interp1d(cdf, xs, fill_value='extrapolate')
    # Perform inverse transform sampling using the interpolant
    >>> r = ppf(rng.uniform(size=1000))

    And compare the histogram:

    >>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()

    c                 ó   — |dkD  |dkD  z  S r�  r‡   )rE   r  r±  s      r6   rc   zstudentized_range_gen._argcheckí/  s   € Ø�A‘˜"˜q™&Ñ!Ð!r8   c                 ó‚   — t        dddt        j                  fd«      }t        dddt        j                  fd«      }||gS )Nr  Fr   r
  r±  r   rh   )rE   r†  r	  s      r6   rk   z!studentized_range_gen._shape_infoð/  s<   € Ü˜˜U Q¬¯© K°Ó@ˆÜ˜˜u q¬"¯&©& k°>ÓBˆØ�CˆyÐr8   c                 ó&   •— t         ‰| �  |d¬«      S )N)rU   r   rM  rS  rT  s     €r6   r•  zstudentized_range_gen._fitstartõ/  s   ø€ ä‰wÑ  ¨FÐ Ó3Ð3r8   c                 óÎ   ‡‡‡— dŠ| j                  «       \  ŠŠˆˆˆfd„}t        j                  |dd«      }t        j                   ||||«      t        j                  ¬«      d   S )NÚ_studentized_range_momentc                 ó°  •— t        j                  ||«      }| |||g}t        j                  |t        «      j
                  j                  t
        j                  «      }t        j                  t         ‰|«      }t        j                   t        j                  fdt        j                  f‰	‰
fg}t        dd¬«      }t        j                  |||¬«      d   S )Nr   r  çê-�™—q=©rš  r™  ©ÚrangesÚopts)r   Ú_studentized_range_pdf_logconstrP   r›  rR  rœ  r�  rž  r   rŸ  ri   Údictr   Únquad)rŒ  r  r±  Ú	log_constÚargÚusr_datar¥  r£  r¤  rÚ  rÙ  Úcython_symbols            €€€r6   Ú_single_momentz3studentized_range_gen._munp.<locals>._single_momentÿ/  s§   ø€ Ü×>Ñ>¸qÀ"ÓEˆIØ�a˜˜YÐ'ˆCÜ—x‘x ¤UÓ+×2Ñ2×:Ñ:¼6¿?¹?ÓKˆHä"×.Ñ.¬v°}ÀhÓOˆCäŸ™�w¤§¡Ð'¨!¬R¯V©V¨°r¸2°hÐ?ˆFÜ˜u¨UÔ3ˆDä—?‘? 3¨v¸DÔAÀ!ÑDÐDr8   r†  r   r  r‡   )r–   rP   Ú
frompyfuncrû   r–  )	rE   rŒ  r  r±  r¬  ÚufuncrÚ  rÙ  r«  s	         @@@r6   r  zstudentized_range_gen._munpù/  sV   ú€ Ø3ˆØ×"Ñ"Ó$‰ˆˆBö
	Eô —‘˜n¨a°Ó3ˆÜ�z‰z™%  1 b›/´·±Ô<¸RÑ@Ð@r8   c                 ó–   — d„ }t        j                  |dd«      }t        j                   ||||«      t         j                  ¬«      d   S )Nc                 óŠ  — |dk  r“d}t        j                  ||«      }| |||g}t        j                  |t        «      j
                  j                  t
        j                  «      }t        j                   t        j                  fdt        j                  fg}nid}| |g}t        j                  |t        «      j
                  j                  t
        j                  «      }t        j                   t        j                  fg}t        j                  t         ||«      }t        dd¬«      }	t        j                  |||	¬«      d   S )	Né † Ú_studentized_range_pdfr   Ú!_studentized_range_pdf_asymptoticr  r   r¡  r¢  )r   r¥  rP   r›  rR  rœ  r�  rž  ri   r   rŸ  r¦  r   r§  ©
r}   r  r±  r«  r¨  r©  rª  r£  r¥  r¤  s
             r6   Ú_single_pdfz/studentized_range_gen._pdf.<locals>._single_pdf0  sû   € ð �FŠ{Ø 8�Ü"×BÑBÀ1ÀbÓI�	Ø˜!˜R Ð+�ÜŸ8™8 C¬Ó/×6Ñ6×>Ñ>¼v¿¹ÓO�ÜŸF™F˜7¤B§F¡FÐ+¨a´·±¨[Ð9‘ð !D�Ø˜!�f�ÜŸ8™8 C¬Ó/×6Ñ6×>Ñ>¼v¿¹ÓO�ÜŸF™F˜7¤B§F¡FÐ+Ð,�ä"×.Ñ.¬v°}ÀhÓOˆCÜ˜u¨UÔ3ˆDÜ—?‘? 3¨v¸DÔAÀ!ÑDÐDr8   r†  r   r  r‡   )rP   r­  rû   r–  )rE   rq   r  r±  rµ  r®  s         r6   rr   zstudentized_range_gen._pdf0  s>   € ò	Eô( —‘˜k¨1¨aÓ0ˆÜ�z‰z™%  1 b›/´·±Ô<¸RÑ@Ð@r8   c           	      óÀ   — d„ }t        j                  |dd«      }t        j                  t        j                   ||||«      t         j                  ¬«      d   dd«      S )Nc                 óŠ  — |dk  r“d}t        j                  ||«      }| |||g}t        j                  |t        «      j
                  j                  t
        j                  «      }t        j                   t        j                  fdt        j                  fg}nid}| |g}t        j                  |t        «      j
                  j                  t
        j                  «      }t        j                   t        j                  fg}t        j                  t         ||«      }t        dd¬«      }	t        j                  |||	¬«      d   S )	Nr±  Ú_studentized_range_cdfr   Ú!_studentized_range_cdf_asymptoticr  r   r¡  r¢  )r   Ú_studentized_range_cdf_logconstrP   r›  rR  rœ  r�  rž  ri   r   rŸ  r¦  r   r§  r´  s
             r6   Ú_single_cdfz/studentized_range_gen._cdf.<locals>._single_cdf)0  sû   € ð
 �FŠ{Ø 8�Ü"×BÑBÀ1ÀbÓI�	Ø˜!˜R Ð+�ÜŸ8™8 C¬Ó/×6Ñ6×>Ñ>¼v¿¹ÓO�ÜŸF™F˜7¤B§F¡FÐ+¨a´·±¨[Ð9‘ð !D�Ø˜!�f�ÜŸ8™8 C¬Ó/×6Ñ6×>Ñ>¼v¿¹ÓO�ÜŸF™F˜7¤B§F¡FÐ+Ð,�ä"×.Ñ.¬v°}ÀhÓOˆCÜ˜u¨UÔ3ˆDÜ—?‘? 3¨v¸DÔAÀ!ÑDÐDr8   r†  r   r  r‡   r   )rP   r­  r�	  rû   r–  )rE   rq   r  r±  r»  r®  s         r6   ru   zstudentized_range_gen._cdf'0  sM   € ò	Eô, —‘˜k¨1¨aÓ0ˆô �w‰w”r—z‘z¡%¨¨1¨b£/¼¿¹ÔDÀRÑHÈ!ÈQÓOÐOr8   )rƒ   r„   r…   r†   rc   rk   r•  r  rr   ru   rÐ  rÑ  s   @r6   r™  r™  ‚/  s+   ø„ ñhòT"òô
4òAò*Aö2Pr8   r™  Ústudentized_range)r�   r‹   rŒ   c                   ó\   ‡ — e Zd ZdZd„ Zd„ Zd„ Zd„ Zd„ Zd„ Z	 e
e«      ˆ fd„«       Zˆ xZS )	Úrel_breitwigner_genaÿ  A relativistic Breit-Wigner random variable.

    %(before_notes)s

    See Also
    --------
    cauchy: Cauchy distribution, also known as the Breit-Wigner distribution.

    Notes
    -----

    The probability density function for `rel_breitwigner` is

    .. math::

        f(x, \rho) = \frac{k}{(x^2 - \rho^2)^2 + \rho^2}

    where

    .. math::
        k = \frac{2\sqrt{2}\rho^2\sqrt{\rho^2 + 1}}
            {\pi\sqrt{\rho^2 + \rho\sqrt{\rho^2 + 1}}}

    The relativistic Breit-Wigner distribution is used in high energy physics
    to model resonances [1]_. It gives the uncertainty in the invariant mass,
    :math:`M` [2]_, of a resonance with characteristic mass :math:`M_0` and
    decay-width :math:`\Gamma`, where :math:`M`, :math:`M_0` and :math:`\Gamma`
    are expressed in natural units. In SciPy's parametrization, the shape
    parameter :math:`\rho` is equal to :math:`M_0/\Gamma` and takes values in
    :math:`(0, \infty)`.

    Equivalently, the relativistic Breit-Wigner distribution is said to give
    the uncertainty in the center-of-mass energy :math:`E_{\text{cm}}`. In
    natural units, the speed of light :math:`c` is equal to 1 and the invariant
    mass :math:`M` is equal to the rest energy :math:`Mc^2`. In the
    center-of-mass frame, the rest energy is equal to the total energy [3]_.

    %(after_notes)s

    :math:`\rho = M/\Gamma` and :math:`\Gamma` is the scale parameter. For
    example, if one seeks to model the :math:`Z^0` boson with :math:`M_0
    \approx 91.1876 \text{ GeV}` and :math:`\Gamma \approx 2.4952\text{ GeV}`
    [4]_ one can set ``rho=91.1876/2.4952`` and ``scale=2.4952``.

    To ensure a physically meaningful result when using the `fit` method, one
    should set ``floc=0`` to fix the location parameter to 0.

    References
    ----------
    .. [1] Relativistic Breit-Wigner distribution, Wikipedia,
           https://en.wikipedia.org/wiki/Relativistic_Breit-Wigner_distribution
    .. [2] Invariant mass, Wikipedia,
           https://en.wikipedia.org/wiki/Invariant_mass
    .. [3] Center-of-momentum frame, Wikipedia,
           https://en.wikipedia.org/wiki/Center-of-momentum_frame
    .. [4] M. Tanabashi et al. (Particle Data Group) Phys. Rev. D 98, 030001 -
           Published 17 August 2018

    %(example)s

    c                 ó   — |dkD  S r2  r‡   ©rE   Úrhos     r6   rc   zrel_breitwigner_gen._argcheck‡0  s   € Ø�Q‰wˆr8   c                 ó@   — t        dddt        j                  fd«      gS )NrÁ  Fr   r
  rh   rj   s    r6   rk   zrel_breitwigner_gen._shape_infoŠ0  rp  r8   c           
      ó6  — t        j                  ddd|dz  z  z   z  dt        j                  dd|dz  z  z   «      z   z  «      dz  t         j                  z  }t        j                  d¬«      5  |||z
  ||z   z  |z  dz  dz   z  cd d d «       S # 1 sw Y   y xY w)NrU   r   r9  rq  )rP   rÿ   rñ   r<  )rE   rq   rÁ  rC  s       r6   rr   zrel_breitwigner_gen._pdf�0  sž   € ä�G‰GØ��Q�s˜A‘v‘X‘Ñ !¤b§g¡g¨a°!°C¸±F±(©lÓ&;Ñ";Ñ<ó
àñä—‘ñˆô �[‰[˜hÔ'ñ 	:Ø˜!˜c™' A¨¡GÑ,¨SÑ0°1Ñ4°qÑ8Ñ9÷	:÷ 	:ò 	:ús   Á.BÂBc           
      ó’  — t        j                  ddt        j                  dd|dz  z  z   «      z   z  «      t         j                  z  }t        j                  dd|z  z   «      t        j                  |t        j                  | |dz   z  «      z  «      z  }|dz  t        j                  |«      z  }t        j
                  |d d«      S )NrU   r   r¿  rG  )rP   rÿ   rñ   rð  Úimagr�	  )rE   rq   rÁ  rC  r  s        r6   ru   zrel_breitwigner_gen._cdf•0  s¥   € ä�G‰G�A�qœ2Ÿ7™7 1 q¨¨a©¡x¡<Ó0Ñ0Ñ1Ó2´2·5±5Ñ8ˆä�G‰G�B˜˜C™‘KÓ Ü�i‰i˜œ"Ÿ'™' 3 $¨¨b©¡/Ó2Ñ2Ó3ñ4ð 	ð �Q‘œŸ™ ›Ñ(ˆä�w‰w�v˜t QÓ'Ð'r8   c                 óR  — |dk(  ry|dk(  rƒt        j                  ddd|dz  z  z   z  dt        j                  dd|dz  z  z   «      z   z  «      t         j                  z  |z  }|t         j                  dz  t        j                  |«      z   z  S |dk(  r…t        j                  dd|dz  z  z   ddt        j                  dd|dz  z  z   «      z   z  z  «      |z  }d|dz  z
  t        j                  dd|z  z
  «      z  }d|z  t        j                  |«      z  S t         j
                  S )Nr   r‰   r   rU   rG  r¿  )rP   rÿ   rñ   rð  rÌ  ri   )rE   rb   rÁ  rC  r  s        r6   r  zrel_breitwigner_gen._munp 0  s  € Ø�Š6ØØ�Š6ä—‘Ø�Q˜˜3 ™6™‘\Ñ" a¬"¯'©'°!°a¸¸Q¹±h±,Ó*?Ñ&?Ñ@óä—‘ñàñˆAð œŸ™˜a™¤"§)¡)¨C£.Ñ0Ñ1Ð1Ø�Š6ä—‘Ø�Q�s˜A‘v‘X‘ ! q¬2¯7©7°1°q¸¸a¹±x±<Ó+@Ñ'@Ñ"AÑBóàñˆAð ˜# ™(‘l¤b§g¡g¨b°2°c±6©kÓ&:Ñ:ˆFØ�q‘5œ2Ÿ7™7 6›?Ñ*Ð*ä—6‘6ˆMr8   c                 óF   — d d t         j                  t         j                  fS rN   r›  rÀ  s     r6   r   zrel_breitwigner_gen._stats³0  s   € ð �Tœ2Ÿ6™6¤2§6¡6Ð)Ð)r8   c                 óœ  •— t        | |||«      \  }}}}t        |t        «      }|r!|j                  «       dk(  r|j                  }d}|�|rt        ‰| �  |g|¢­i |¤ŽS |€8t        j                  ||z
  g d¢«      \  }}	}
|
|z
  }|	|z  }|s|g}d|vr(||d<   n"t        j                  ||z
  «      }||z  }|s|g}t        ‰| �  |g|¢­i |¤ŽS )Nr   F)rÕ  r”   g      è?r/   )
rG  r?   r*   r@   rD   rA   rC   rP   Úquantiler�  )rE   rF   rG   r5   r	  rö   r÷   rH   r¨  r©  rª  Úscale_0Úrho_0ÚM_0r–  s                 €r6   rC   zrel_breitwigner_gen.fit¹0  s  ø€ ô !<Ø�$˜˜dó!
Ñˆˆa��vô ˜d¤LÓ1ˆÙØ× Ñ Ó" aÒ'ð ×'Ñ'�Ø �àˆ<™8Ü‘7‘;˜tÐ3 dÒ3¨dÑ3Ð3àˆ>ô ŸK™K¨¨t©Ò5FÓG‰MˆC��cØ˜C‘iˆGØ˜'‘MˆEÙØ�w�Ø˜dÑ"Ø '��W’ä—)‘)˜D 4™KÓ(ˆCØ˜&‘LˆEÙØ�w�Ü‰w‰{˜4Ð/ $Ò/¨$Ñ/Ð/r8   )rƒ   r„   r…   r†   rc   rk   rr   ru   r  r   r   r   rC   rÐ  rÑ  s   @r6   r¾  r¾  I0  sA   ø„ ñ<òzòGò:ò	(òò&*ñ ˜MÓ*ó 0ó +ô 0r8   r¾  Úrel_breitwignerrN   (Q  rH  Úcollections.abcr   Ú	functoolsr   r   rœ  ÚnumpyrP   Únumpy.polynomialr   Úscipy.interpolater   Úscipy._lib.doccerr	   r
   r   Úscipy._lib._ccallbackr   Úscipyr   r   Úscipy.specialÚspecialrw   Úscipy.special._ufuncsrÐ  rn   Úscipy._lib._utilr   r   rU  r   Ú_tukeylambda_statsr   rÍ  r   rÎ  Ú_distn_infrastructurer   r   r   r   r   r   r   r   r   Ú(scipy.stats._distribution_infrastructurer   Ú_ksstatsr   r    r!   Ú
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