An overview of multivariate gamma distributions as seen from a (multivariate) matrix exponential perspective

Abstract

Numerous definitions of multivariate exponential and gamma distributions can be retrieved from the literature [4]. These distribtuions belong to the class of Multivariate Matrix-- Exponetial Distributions (MVME) whenever their joint Laplace transform is a rational function. The majority of these distributions further belongs to an important subclass of MVME distributions [5, 1] where the multivariate random vector can be interpreted as a number of simultaneously collected rewards during sojourns in a the states of a Markov chain with one absorbing state, the rest of the states being transient. We present the corresponding representations for all such distributions. In this way we obtain a unification of the variety of existing distributions as well as a deeper understanding of their probabilistic nature and a clarification of their similarities and differences. In particular one may easily generalize or combine any of the known distributions by modifying the generators adequately. Also, it is straightforward to simulate from this class. Thus, by identifying distributions as belonging to this subclass it becomes apparent how to simulate from most previously discussed distributions with rational Laplace transform. In a longer perspective stochastic and statistical analysis for MVME will in particular apply to any of the previously defined distributions. Multivariate gamma distributions have been used in a variety of fields like hydrology, [11], [10], [6], space (wind modeling) [9] reliability [3], [7], traffic modeling [8], and, finance [2]. It is our hope that our the paper will assist practitioners in formulating and analyzing models in a much more transparent and easily accessible way.