Power Log Normal Distribution#
A generalization of the log-normal distribution with shape parameters \(\sigma>0\), \(c>0\) and support \(x\geq0\).
\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*}
This distribution reduces to the log-normal distribution when \(c=1.\)
Implementation: scipy.stats.powerlognorm