Negative multinomial distribution

Abstract

This paper reviews properties of the negative multinomial distribution and some related distributions. On the negative binomial distribution (NBn) much has been wri t ten and the contributions were summarized in two recent survey reports ([3], [9]). In the course of researches on the NBn its multivariate extension has been tried. The notion of the negative multinomial distribution (NMn) was first introduced in the model of the inverse sampling in multiple Bernoulli trials and accordingly the parameter " k " was limited to integral values. Later, in the papers discussing statistical theory of accident, absenteeism and contagion, the NMn was introduced under the name " multivariate negative binomial d i s t r ibu t ion" ([1], [2], [4], etc.). Among them Bates and Neyman's paper [4] was the first which t reated the NMn systematically. Surveying the properties of the NMn we remark the relations among distributions, which make clear the probabilistic s t ructure of the individual distributions. We notice especially tha t the relation between the binomial distribution (Bn) and the NBn is quite similar to tha t between the multinomial distribution (Mn) and the NMn, so the name NMn is preferable to the multivariate NBn. On the way of discussions a multivariate extension of Fisher 's logarithmic series, the negative hypergeometric distribution (NHg) and its multivariate extension will be treated. Here, the name NHg is proposed, though this distribution has been discussed in literatures under different names.