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1997 | OriginalPaper | Buchkapitel

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

verfasst von : Gregory A. Tripsiannis, Andreas N. Philippou

Erschienen in: Advances in Combinatorial Methods and Applications to Probability and Statistics

Verlag: Birkhäuser Boston

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Ling (1989) introduced and studied a negative binomial distribution of order k, type III, which he denoted by NBk 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 $$\overline {MNB} _{k,III} (r;q_1 \ldots ,q_m ),$$.

Metadaten
Titel
A Multivariate Negative Binomial Distribution of Order k Arising When Success Runs are Allowed to Overlap
verfasst von
Gregory A. Tripsiannis
Andreas N. Philippou
Copyright-Jahr
1997
Verlag
Birkhäuser Boston
DOI
https://doi.org/10.1007/978-1-4612-4140-9_25