| Abstract: | Abstract The study of multivariate distributions of order k, two of which are the multivariate negative binomial of order k and the multinomial of the same order, was introduced in Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1988 Philippou, A. N., Antzoulakos, D. L. and Tripsiannis, G. A. 1988. Multivariate distributions of order k. Statistics and Probability Letters, 7(3): 207–216. [Google Scholar]). Multivariate distributions of order k. Statistics and Probability Letters 7(3):207–216.), and Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1990 Philippou, A. N., Antzoulakos, D. L. and Tripsiannis, G. A. 1990. Multivariate distributions of order k, part II. Statistics and Probability Letters, 10(1): 29–35. [Google Scholar]). Multivariate distributions of order k, part II. Statistics and Probability Letters 10(1):29–35.). Recently, an order k (or cluster) generalized negative binomial distribution and a multivariate negative binomial distribution were derived in Sen and Jain (Sen, K., Jain, R. (1996 Sen, K. and Jain, R. 1996. “Cluster generalized negative binomial distribution”. In Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday Edited by: Borthakur, A. C. 227–241. New Delhi: New Age International Publishers. [Google Scholar]). Cluster generalized negative binomial distribution. In: Borthakur et al. A. C., Eds.; Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday. New Age International Publishers: New Delhi, 227–241.) and Sen and Jain (Sen, K., Jain, R. (1997 Sen, K. and Jain, R. 1997. A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics—Theory and Methods, 26: 871–884. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics-Theory and Methods 26:871–884.), respectively. In this paper, all four distributions are generalized to a multivariate generalized negative binomial distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases several known and new multivariate distributions of order k, and gives rise in the limit to multivariate generalized logarithmic, Poisson and Borel-Tanner distributions of the same order. Applications are indicated. |
