.. _continuous-dpareto_lognorm:

Double Pareto Lognormal Distribution

For real numbers :math:x and :math:\mu, :math:\sigma > 0, :math:\alpha > 0, and :math:\beta > 0, the PDF of a double Pareto lognormal distribution is:

.. math:: :nowrap:

\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 :math:R(t) = \frac{1 - \Phi(t)}{\phi(t)} is a Mills' ratio, :math:y_1 = \alpha \sigma - \frac{\log x - \mu}{\sigma}, and :math:y_2 = \beta \sigma + \frac{\log x - \mu}{\sigma}. The CDF is:

.. math:: :nowrap:

\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 :math:k > \alpha is given by:

.. math:: :nowrap:

\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