幂对数正态分布#
对数正态分布的推广，具有形状参数 \(\sigma>0\)，\(c>0\) 以及支持 \(x\geq0\).
\begin{eqnarray*} f\left(x;\sigma,c\right) & = & \frac{c}{x\sigma}\phi\left(\frac{\log x}{\sigma}\right)\left(\Phi\left(-\frac{\log x}{\sigma}\right)\right)^{c-1}\\ F\left(x;\sigma,c\right) & = & 1-\left(\Phi\left(-\frac{\log x}{\sigma}\right)\right)^{c}\\ G\left(q;\sigma,c\right) & = & \exp\left(-\sigma\Phi^{-1}\left(\left(1-q\right)^{1/c}\right)\right)\end{eqnarray*}
\[\mu_{n}^{\prime}=\int_{0}^{1}\exp\left(-n\sigma\Phi^{-1}\left(y^{1/c}\right)\right)dy\]
\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*}
当 \(c=1.\) 时，此分布简化为对数正态分布。