Multivariate negative binomial distributions generated by multivariate exponential distributions

Abstract

We define a multivariate negative binomial distribution (MVNB) as a bivariate Poisson distribution function mixed with a mul-tivariate exponential (MVE) distribution. We focus on the class of MVNB distributions generated by Marshall–Olkin MVE distributions. For simplic-ity of notation we analyze in detail the class of bivariate (BVNB) distribu-tions. In applications the standard data from [2] and [7] and data concerning parasites of birds from [4] are used. 1. Introduction. It is known that a univariate geometrical probability distribution function is a mixed Poisson distribution with exponentially dis-tributed parameter. A univariate negative binomial distribution is a mixed Poisson distribution where the mixing parameter has a gamma distribution. Also it is easy to see, considering convolution and mixture, that mutually corresponding are: the class of negative binomial distributions and the class of gamma distributions. These univariate properties suggest the definition of a multivariate negative binomial (MVNB) distribution on the basis of multivariate exponential (MVE) distributions and convolution. There exist a few variants of MVE distributions with exponential marginals; we focus on the class of Marshall–Olkin MVE distributions. For simplicity of notation we consider in detail bivariate (BVNB) distributions defined by the BVE class. We present three applications of BVNB distributions using the standard data from [2] and [7] concerning accidents, and the data concerning the number of parasites of the pheasant [4]. A new variant of MVNB