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Generalized Gamma Distribution

A general probability form that reduces to many common distributions: $x>0$ $a>0$ and $c\neq0.$

$$\begin{eqnarray*} f\left(x;a,c\right) & = & \frac{\left|c\right|x^{ca-1}}{\Gamma\left(a\right)}\exp\left(-x^{c}\right)\\ F\left(x;a,c\right) & = & \begin{array}{cc} \frac{\Gamma\left(a,x^{c}\right)}{\Gamma\left(a\right)} & c>0\\ 1-\frac{\Gamma\left(a,x^{c}\right)}{\Gamma\left(a\right)} & c<0\end{array}\\ G\left(q;a,c\right) & = & \left\{ \Gamma^{-1}\left[a,\Gamma\left(a\right)q\right]\right\} ^{1/c}\quad c>0\\ & & \left\{ \Gamma^{-1}\left[a,\Gamma\left(a\right)\left(1-q\right)\right]\right\} ^{1/c}\quad c<0\end{eqnarray*}$$

$$\mu_{n}^{\prime}=\frac{\Gamma\left(a+\frac{n}{c}\right)}{\Gamma\left(a\right)}$$

$$\begin{eqnarray*} \mu & = & \frac{\Gamma\left(a+\frac{1}{c}\right)}{\Gamma\left(a\right)}\\ \mu_{2} & = & \frac{\Gamma\left(a+\frac{2}{c}\right)}{\Gamma\left(a\right)}-\mu^{2}\\ \gamma_{1} & = & \frac{\Gamma\left(a+\frac{3}{c}\right)/\Gamma\left(a\right)-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\ \gamma_{2} & = & \frac{\Gamma\left(a+\frac{4}{c}\right)/\Gamma\left(a\right)-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\\ m_{d} & = & \left(\frac{ac-1}{c}\right)^{1/c}.\end{eqnarray*}$$

Special cases are Weibull $\left(a=1\right)$ , half-normal $\left(a=1/2,c=2\right)$ and ordinary gamma distributions $c=1.$ If $c=-1$ then it is the inverted gamma distribution.

$$h\left[X\right]=a-a\Psi\left(a\right)+\frac{1}{c}\Psi\left(a\right)+\log\Gamma\left(a\right)-\log\left|c\right|.$$

Implementation: scipy.stats.gengamma