Double Pareto Lognormal Distribution#
For real numbers \(x\) 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