Asymmetric Laplace Distributions

Abstract

Chapter 3 is devoted to asymmetric Laplace distributions — a skewed family of distributions that in our opinion is the most appropriate skewed generalization of the classical Laplace law. In the last several decades, various forms of skewed Laplace distributions have sporadically appeared in the literature. One of the earliest is due to McGill (1962), who considers distributions with p.d.f.$$ f(x) = \left\{ {\begin{array}{*{20}c} {\frac{{\varphi _1 }} {2}e^{ - \varphi _1 |x - \theta |} , x \leqslant \theta ,} \\ {\frac{{\varphi _2 }} {2}e^{ - \varphi _2 |x - \theta |} , x > \theta ,} \\ \end{array} } \right. $$ (3.0.1) while Holla and Bhattacharya (1968) study the distribution with p.d.f. $$ f(x) = \left\{ {\begin{array}{*{20}c} {p\varphi e^{ - \varphi \left| {x - \theta } \right|} , x \leqslant \theta ,} \\ {(1 - p)\varphi e^{ - \varphi \left| {x - \theta } \right|} , \theta < x,} \\ \end{array} } \right. $$ (3.0.2) where 0 < p < 1. Lingappaiah (1988) derived some properties of (3.0.1), terming the distribution two-piece double exponential. Poiraud-Casanova and Thomas-Agnan (2000) exploited a skewed Laplace distribution with p.d.f. $$f\left( x \right) = \alpha \left( {1 - \alpha } \right)\left\{ {\begin{array}{*{20}{c}} {{{e}^{{ - \left( {1 - \alpha } \right)\left| {x - \theta } \right|}}}, for x < \theta ,} \hfill \\ {{{e}^{{ - \alpha \left| {x - \theta } \right|}}}, for x \geqslant \theta ,} \hfill \\ \end{array} } \right. $$ (3.0.3) and α∈(0,1), to show the equivalence of certain quantile estimators.