对数伽马分布¶
单个形状参数 \(c>0\) 。支持是 \(x\in\mathbb{{R}}\) 。
\Begin{eqnarray*}f\Left(x；c\Right)&=&\frac{\exp\left(cx-e^{x}\right)}{\Gamma\left(c\right)}\\
F\Left(x；c\Right)&=&\frac{\Gamma\Left(c，e^{x}\Right)}{\Gamma\Left(c\Right)}\\
g\Left(q；c\Right)&=&\LOG\Left(\Gamma^{-1}\Left(c，q\Gamma\left(c\right)\end{eqnarray*}
哪里 \(\gamma\) 是较低的不完全伽马函数， \(\gamma\left(s, x\right) = \int_0^x t^{{s-1}} e^{{-t}} dt\) 。
\[\mu_{n}^{\prime}=\int_{0}^{\infty}\left [\log y\right] ^{n}y^{c-1}\exp\Left(-y\Right)dy。\]
\BEGIN{eqnarray*}\MU&=&\MU_{1}^{\PRIME}\\
\MU_{2}&=&\MU_{2}^{\PRIME}-\MU^{2}\\
\Gamma_{1}&=&\frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\Gamma_{2}&=&\frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}