Asymmetric Multivariate Laplace Distribution

In this chapter we present the theory of a class of multivariate laws that we term asymmetric Laplace (AL) distributions [see Kozubowski and Podgórski (1999bc), Kotz et al. (2000b)]. The class is an extension of both the symmetric multivariate Laplace distributions and the univariate AL distributions that were discussed in previous chapters. This extension retains the natural, asymmetric, and multivariate features of the properties characterizing these two important subclasses. In particular, the AL distributions arise as the limiting laws in a random summation scheme with i.i.d. terms having a finite second moment, where the number of terms in the summation is geometrically distributed independently of the terms themselves. This class can be viewed as a subclass of hyperbolic distributions and some of its properties are inherited from them. However, to demonstrate an elegant theoretical structure of the multivariate AL laws and also for the sake of simplicity we prefer direct derivations of the results. Thus we provide explicit formulas for the probability density and the density of the Lévy measure. The results presented also include characterizations, mixture representations, formulas for moments, a simulation algorithm, and a brief discussion of linear regression models with AL errors.