Asymmetric Laplace Distribution

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

Functions

\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*}

\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