广义逻辑分布#
已用于极值分析。有一个形状参数 \(c>0\)。支持为 \(x \in \mathcal{R}\).
\begin{eqnarray*} f\left(x;c\right) & = & \frac{c\exp\left(-x\right)}{\left[1+\exp\left(-x\right)\right]^{c+1}}\\ F\left(x;c\right) & = & \frac{1}{\left[1+\exp\left(-x\right)\right]^{c}}\\ G\left(q;c\right) & = & -\log\left(q^{-1/c}-1\right)\end{eqnarray*}
\[M\left(t\right)=\frac{c}{1-t}\,_{2}F_{1}\left(1+c,\,1-t\,;\,2-t\,;-1\right)\]
\begin{eqnarray*} \mu & = & \gamma+\psi_{0}\left(c\right)\\ \mu_{2} & = & \frac{\pi^{2}}{6}+\psi_{1}\left(c\right)\\ \gamma_{1} & = & \frac{\psi_{2}\left(c\right)+2\zeta\left(3\right)}{\mu_{2}^{3/2}}\\ \gamma_{2} & = & \frac{\left(\frac{\pi^{4}}{15}+\psi_{3}\left(c\right)\right)}{\mu_{2}^{2}}\\ m_{d} & = & \log c\\ m_{n} & = & -\log\left(2^{1/c}-1\right)\end{eqnarray*}
请注意，多伽马函数为
\begin{eqnarray*} \psi_{n}\left(z\right) & = & \frac{d^{n+1}}{dz^{n+1}}\log\Gamma\left(z\right)\\ & = & \left(-1\right)^{n+1}n!\sum_{k=0}^{\infty}\frac{1}{\left(z+k\right)^{n+1}}\\ & = & \left(-1\right)^{n+1}n!\zeta\left(n+1,z\right)\end{eqnarray*}
其中 \(\zeta\left(k,x\right)\) 是黎曼 zeta 函数的推广，称为赫维茨 zeta 函数。请注意 \(\zeta\left(n\right)\equiv\zeta\left(n,1\right)\).