Double Pareto Lognormal Distribution#
For real numbers and \mu, \sigma > 0, \alpha > 0, and \beta > 0, the PDF of a double Pareto lognormal distribution is:
\begin{eqnarray*}
f(x, \mu, \sigma, \alpha, \beta) =
\frac{\alpha \beta}{(\alpha + \beta) x}
\phi\left( \frac{\log x - \mu}{\sigma} \right)
\left( R(y_1) + R(y_2) \right)
\end{eqnarray*}
where R(t) = \frac{1 - \Phi(t)}{\phi(t)} is a Mills’ ratio, y_1 = \alpha \sigma - \frac{\log x - \mu}{\sigma}, and y_2 = \beta \sigma + \frac{\log x - \mu}{\sigma}. The CDF is:
\begin{eqnarray*}
F(x, \mu, \sigma, \alpha, \beta) =
\Phi \left(\frac{\log x - \mu}{\sigma} \right) -
\phi \left(\frac{\log x - \mu}{\sigma} \right)
\left(\frac{\beta R(x_1) - \alpha R(x_2)}{\alpha + \beta} \right)
\end{eqnarray*}
Raw moment k > \alpha is given by:
\begin{eqnarray*}
\mu_k' = \frac{\alpha \beta}{(\alpha - k)(\beta + k)}
\exp \left(k \mu + \frac{k^2 \sigma^2}{2} \right)
\end{eqnarray*}
Implementation: scipy.stats.dpareto_lognorm