Inverted Gamma Distribution

Special case of the generalized Gamma distribution with \(c=-1\) and \(a>0\) and support \(x\geq0\).

\begin{eqnarray*} f\left(x;a\right) & = & \frac{x^{-a-1}}{\Gamma\left(a\right)}\exp\left(-\frac{1}{x}\right)\\ F\left(x;a\right) & = & \frac{\Gamma\left(a,\frac{1}{x}\right)}{\Gamma\left(a\right)}\\ G\left(q;a\right) & = & \left\{ \Gamma^{-1}\left(a,\Gamma\left(a\right)q\right)\right\} ^{-1}\end{eqnarray*}

\[\mu_{n}^{\prime}=\frac{\Gamma\left(a-n\right)}{\Gamma\left(a\right)}\quad a>n\]

\begin{eqnarray*} \mu & = & \frac{1}{a-1}\quad a>1\\ \mu_{2} & = & \frac{1}{\left(a-2\right)\left(a-1\right)}-\mu^{2}\quad a>2\\ \gamma_{1} & = & \frac{\frac{1}{\left(a-3\right)\left(a-2\right)\left(a-1\right)}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\ \gamma_{2} & = & \frac{\frac{1}{\left(a-4\right)\left(a-3\right)\left(a-2\right)\left(a-1\right)}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}

\[m_{d}=\frac{1}{a+1}\]

\[h\left[X\right]=a-\left(a+1\right)\psi\left(a\right)+\log\Gamma\left(a\right).\]

where \(\Psi\) is the digamma function \(\psi(z) = \frac{d}{dz} \log(\Gamma(z))\).

Implementation: scipy.stats.invgamma