幂对数正态分布¶
带形状参数的对数正态分布的推广 \(\sigma>0\) ， \(c>0\) 和支持 \(x\geq0\) 。
\BEGIN{eqnarray*}f\Left(x；\sigma，c\right)&=&\frac{c}{x\sigma}\Phi\Left(\frac{\logx}{\sigma}\right)\Left(\Phi\Left(-\frac{\logx}{\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*}
在以下情况下，此分布减小为对数正态分布 \(c=1.\)