Compound negative binomial distribution with negative multinomial summands

Abstract

The class of Negative Binomial distributions seems to be introduced by Greenwood and Yule in 1920. Due to its wide spread application, investigations of distributions, closely related with it will be always contemporary. Bates, Neyman and Wishart introduce Negative Multinomial distribution. They reach it considering the mixture of independent Poisson distributed random variables with one and the same Gamma mixing variable.This paper investigates a particular case of multivariate compound distribution with one and the same compounding variable. In our case it is Negative Binomial or Sifted Negative Binomial. The summands with equal indexes in different coordinates are Negative Multinomially distributed. In case without shifting, considered as a mixture, the resulting distribution coincides with Mixed Negative Multinomial distribution with scale changed Negative Binomially distributed first parameter. We prove prove that it is Multivariate Power Series Distributed and find explicit form of its parameters. When the summands are geometrically distributed this distribution is stochastically equivalent to a product of independent Bernoulli random variable and appropriate multivariate Geometrically distributed random vector. We show that Compound Shifted Negative Binomial Distribution with Geometric Summands is a particular case of Negative Multinomial distribution with new parameters.