Multivariate Generalized Distributions of Order k

• Published in: Communications in statistics. Theory and methods Vol. 32; no. 9; pp. 1725 - 1735
• Main Authors: Tripsiannis, Gregory A., Philippou, Andreas N., Papathanasiou, Afroditi A.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 08.01.2003; Taylor & Francis
• Subjects: AMS 2000 Subject Classifications: 62E15; Binomial-delta; Borel-Tanner; Combinatorial probability; Distribution theory; Exact sciences and technology; Generalized; Haight; Limiting cases; Logarithmic series; Mathematics; Mean; Multinomial; Multivariate analysis; Multivariate distributions of order k; Negative binomial; Negative binomial-delta; Poisson; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Statistics; Takács; Variance-covariance; Delhi; Asia; India; Data analysis; Statistical distribution; Cluster analysis (statistics); Borel Tanner distribution; Generalized function; 1990; Multivariate distribution; Probability distribution; Limiting case; Multivariate analysis; Poisson distribution; Negative binomial distribution; Statistical method; 1997; 1996; Distribution function; Binomial distribution; Multinomial distribution; Sampling; 1988
• ISSN: 0361-0926; 1532-415X
• DOI: 10.1081/STA-120022705

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 ). 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 ). 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 ). 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 ). 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.