On the compound negative multinomial distribution and correlations among inversely sampled pollen counts

Abstract

INTRODUCTION In the generalization of Bernoulli trials where we have k possible outcomes of each trial, k let the probability of the ith outcome in each trial be pi (i = 1,...,k) where Pi= 1. i=l For a fixed number of trials (n), the probability of exactly x1 occurrences of outcome 1, x2 of 2, ..., Xk of k is given by the multinomial distribution. If, however, the number of trials is not fixed in advance, but we continue to consider new trials until exactly r occurrences of the kth outcome (say) have been noted, then the probability of exactly xl occurrences of outcome 1, x2 of 2, ..., Xk1 of k 1, is given by the negative multinomial distribution. This latter procedure we shall refer to as 'inverse sampling' (cf. Tweedie, 1952). The negative multinomial distribution is clearly a generalization of the negative binomial. Just as this latter can be deduced from widely different models, so also can the negative multinomial. Thus Bates & Neyman (1952) arrive at this distribution (their multivariate negative binomial distribution) by considering the joint distribution of kIc-1 independent Poisson variables Xi (i = il ..., Ik-i) each with mean ai A. The parameter A is in turn regarded as a random variable A having a y(or type III) distribution. Integrating the joint frequency function of the X's and A with respect to the latter reveals that the marginal frequency function of the X's is that of the negative multinomial. (Using this approach r (r > 0) is the parameter of a y-distribution and is not restricted to integer values as in the model considered here.) Wishart (1949, p. 48) uses the term 'Pascal multinomial' for the case where r is an integer. The terms of the negative multinomial distribution are readily obtained as the coefficients k-1 of II t in the expansion of k-1 ]