.. _continuous-laplace-asymmetric:

Asymmetric Laplace Distribution

This distribution is a generalization of the Laplace distribution. It has a single shape parameter :math:\kappa>0 that species the distribution's asymmetry. The special case :math:\kappa=1 yields the Laplace distribution.

Functions

.. math:: :nowrap:

\begin{eqnarray*}
F(x, \kappa) & = & 1-\frac{\kappa^{-1}}{\kappa+\kappa^{-1}}\exp(-x\kappa),\quad x\ge0; \\
             & = & \frac{\kappa}{\kappa+\kappa^{-1}}\exp(x/\kappa),\quad x<0. \\
f(x, \kappa) & = & \frac{1}{\kappa+\kappa^{-1}}\exp(-x\kappa),\quad x\ge0; \\
             & = & \frac{1}{\kappa+\kappa^{-1}}\exp(x/\kappa),\quad x<0.
\end{eqnarray*}

.. math:: :nowrap:

\begin{eqnarray*}
\mu & = & \kappa^{-1}-\kappa\\
\mu_2 & = & \kappa^{-2}+\kappa^2\\
\gamma_1 & = & \frac{2(1-\kappa^6)}{(1+\kappa^4)^{3/2}}\\
\gamma_2 & = & \frac{6(1+\kappa^8)}{(1+\kappa^4)^2}
\end{eqnarray*}

References

• "Asymmetric Laplace distribution", Wikipedia https://en.wikipedia.org/wiki/Asymmetric_Laplace_distribution

• Kozubowski TJ and Podgórski K, "A Multivariate and Asymmetric Generalization of Laplace Distribution," Computational Statistics 15, 531--540 (2000). :doi:10.1007/PL00022717

Implementation: scipy.stats.laplace_asymmetric