广义Logistic分布¶
已用于极值分析。有一个形状参数 \(c>0.\) 支持是 \(x\geq0\) 。
\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\右)\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*}
哪里 \(\zeta\left(k,x\right)\) 是Riemann Zeta函数的推广，称为Hurwitz Zeta函数。请注意， \(\zeta\left(n\right)\equiv\zeta\left(n,1\right)\) 。