广义伽玛分布#
一种通用的概率形式，可简化为许多常见的分布。它有两个形状参数 \(a>0\) 和 \(c\neq0\)。其支持范围是 \(x\geq0\)。
\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) & = & \left\{ \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} \right. \\ G\left(q;a,c\right) & = & \left\{ \begin{array}{cc} \gamma^{-1} \left(a, \Gamma\left(a\right) q \right)^{1/c} & c>0 \\ \gamma^{-1} \left(a, \Gamma\left(a\right) \left(1-q\right) \right)^{1/c} & c<0 \end{array} \right. \end{eqnarray*}
其中 \(\gamma\) 是下不完全伽玛函数，\(\gamma\left(s, x\right) = \int_0^x t^{s-1} e^{-t} dt\)。
\begin{eqnarray*} \mu_{n}^{\prime} & = & \frac{\Gamma\left(a+\frac{n}{c}\right)}{\Gamma\left(a\right)}\\ \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*}
特殊情况是威布尔分布 \(\left(a=1\right)\)，半正态分布 \(\left(a=1/2,c=2\right)\) 和普通伽玛分布 \(c=1.\) 如果 \(c=-1\)，则它为反伽玛分布。
\[h\left[X\right]=a-a\Psi\left(a\right)+\frac{1}{c}\Psi\left(a\right)+\log\Gamma\left(a\right)-\log\left|c\right|.\]