Double Pareto Lognormal Distribution

For \(x > 0\), real \(\mu\), \(\sigma > 0\), \(\alpha > 0\), 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 \(\phi\) and \(\Phi\) are the standard normal PDF and CDF, respectively, \(R(t) = \frac{1 - \Phi(t)}{\phi(t)}\) 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(y_1) - \alpha R(y_2)}{\alpha + \beta} \right) \end{eqnarray*}

The raw moment of order \(k\), for \(-\beta < k < \alpha\), 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