A Multivariate Negative Binomial Distribution of Order k Arising When Success Runs are Allowed to Overlap

Abstract

Ling (1989) introduced and studied a negative binomial distribution of order k, type III, which he denoted by NB k III(r, p), as the probability distribution of the number of Bernoulli trials M r (k) until the occurrence of r possibly overlapping success runs of length k [see also Hirano et al. (1991)]. In the present paper, independent trials are considered with m + 1 possible outcomes and the multivariate negative binomial distribution of order k, type III, say \(\overline {MNB} _{k,III} (r;q_1 \ldots ,q_m ),\) is introduced as the distribution of a random vector Y which is a multivariate analogue of Y r (k) − (k + r − 1). The probability generating function, mean and variance-covariance, and several distributional properties of Y are established. The present paper generalizes to the multivariate case shifted versions of results of Ling (1989) and Hirano et al. (1991) on NB k,III (r, p). Three new results on NB k,III (r, p) or/and its shifted version are derived first; another one arises as a corollary of a proposition on \(\overline {MNB} _{k,III} (r;q_1 \ldots ,q_m ),\).