doc/source/tutorial/stats/continuous_loggamma.rst
.. _continuous-loggamma:
A single shape parameter :math:c>0 . The support is :math:x\in\mathbb{R}.
.. math:: :nowrap:
\begin{eqnarray*} f\left(x;c\right) & = & \frac{\exp\left(cx-e^{x}\right)}{\Gamma\left(c\right)}\\
F\left(x;c\right) & = & \frac{\gamma\left(c,e^{x}\right)}{\Gamma\left(c\right)}\\
G\left(q;c\right) & = & \log\left(\gamma^{-1}\left(c,q\Gamma\left(c\right)\right)\right)\end{eqnarray*}
where :math:\gamma is the lower incomplete gamma function, :math:\gamma\left(s, x\right) = \int_0^x t^{s-1} e^{-t} dt.
.. math::
\mu_{n}^{\prime}=\int_{0}^{\infty}\left[\log y\right]^{n}y^{c-1}\exp\left(-y\right)dy.
.. math:: :nowrap:
\begin{eqnarray*} \mu & = & \mu_{1}^{\prime}\\
\mu_{2} & = & \mu_{2}^{\prime}-\mu^{2}\\
\gamma_{1} & = & \frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
Implementation: scipy.stats.loggamma