广义伽玛分布¶
一种简化为许多常见分布的一般概率形式。有两个形状参数 \(a>0\) 和 \(c\neq0\) 。支持是 \(x\geq0\) 。
\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)&=&
\左\{
\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{数组}
\对。\\
g\Left(q；a，c\right)&=&
\左\{
\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{数组}
\对。\end{eqnarray*}
哪里 \(\gamma\) 是较低的不完全伽马函数， \(\gamma\left(s, x\right) = \int_0^x t^{{s-1}} e^{{-t}} dt\) 。
\BEGIN{等式*}\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*}
特例是威布尔 \(\left(a=1\right)\) ，半正常 \(\left(a=1/2,c=2\right)\) 和普通的伽马分布 \(c=1.\) 如果 \(c=-1\) 那么它就是倒伽马分布。
\[H\Left [X\right] =a-a\Psi\left(a\right)+\frac{1}{c}\Psi\left(a\right)+\log\Gamma\left(a\right)-\log\left|c\right|.\]