Definition and characterization of multivariate negative binomial distribution

AbstractThe probability generating function (pgf) of an n-variate negative binomial distribution is defined to be [β(s1,…,sn)]−k where β is a polynomial of degree n being linear in each si and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h(x1,…,xn)exp(Σi=1n θixi)f(θ1,…,θn) is negative binomial if and only if its univariate marginals are negative binomial. Let St, t = 1,…, m, be subsets of {s1,…, sn} with empty ∩t=1mSt. Then an n-variate pgf is of a negative binomial if and only if for all s in St being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t