倒伽马分布¶
广义Gamma分布的特例 \(c=-1\) 和 \(a>0\) 和支持 \(x\geq0\) 。
\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*}
\[m_{d}=\frac{1}{a+1}\]
\[H\Left [X\right] =a-\left(a+1\right)\psi\left(a\right)+\log\Gamma\left(a\right).\]
哪里 \(\Psi\) 是Digamma功能 \(\psi(z) = \frac{{d}}{{dz}} \log(\Gamma(z))\) 。