Multivariate distributions of order k. (English) Zbl 0678.62058
Summary: Three multivariate distributions of order k are introduced and studied. A multivariate negative binomial distribution of order k is derived first, by means of an urn scheme, and two limiting cases of it are obtained next. They are, respectively, a multivariate Poisson distribution of order k and a multivariate logarithmic series distribution of the same order. The probability generating functions, means, variances and covariances of these distributions are obtained, and some further genesis schemes of them and interrelationships among them are also established.
The present paper extends to the multivariate case the work of the first author [Ann. Inst. Stat. Math. 40, No.3, 467-475 (1988)] on multiparameter distributions of order k. At the same time, several results of S. Aki [ibid. 37, 205-224 (1985; Zbl 0577.62013)] on extended distributions of order k are also generalized to the multivariate case.
The present paper extends to the multivariate case the work of the first author [Ann. Inst. Stat. Math. 40, No.3, 467-475 (1988)] on multiparameter distributions of order k. At the same time, several results of S. Aki [ibid. 37, 205-224 (1985; Zbl 0577.62013)] on extended distributions of order k are also generalized to the multivariate case.
MSC:
| 62H10 | Multivariate distribution of statistics |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
Keywords:
multivariate distributions of order k; multivariate negative binomial distribution; urn scheme; multivariate Poisson distribution; multivariate logarithmic series distribution; probability generating functions; means; variances; covariances; genesis schemes; interrelationshipsCitations:
Zbl 0577.62013References:
| [1] | Aki, S., Discrete distributions of order κ on a binary sequence, Annals of the Institute of Statistical Mathematics, 37A, 205-224 (1985) · Zbl 0577.62013 |
| [2] | Aki, S.; Kuboki, H.; Hirano, K., On discrete distributions of order κ, Annals of the Institute of Statistical Mathematics, 36A, 431-440 (1984) · Zbl 0572.62018 |
| [3] | Feller, W., (An Introduction to Probability Theory and its Applications, Vol. I (1968), Wiley: Wiley New York) · Zbl 0155.23101 |
| [4] | Johnson, N. L.; Kotz, S., (Discrete Distributions, Vol. 1 (1969), Houghton-Mifflin: Houghton-Mifflin New York) · Zbl 0292.62009 |
| [5] | Hirano, K.; Aki, S., Properties of the extended distributions of order κ, Statistics & Probability Letters, 6, 67-69 (1987) · Zbl 0634.60014 |
| [6] | Patil, G. P.; Bildikar, Sheela, Multivariate logarithmic series distribution as a probability model in population and community ecology and some of its statistical properties, Journal of the American Statistical Association, 62, 655-674 (1967) · Zbl 0171.19103 |
| [7] | Patil, G. P.; Boswell, M. T.; Jochi, S. W.; Ratnaparkhi, M. V., (Dictionary and Classified Bibliography of Statistical Distributions in Scientific Work, Vol. 1 (1984), International Cooperative Publishing House: International Cooperative Publishing House Fairland) |
| [8] | Philippou, A. N., Poisson and compound Poisson distributions of order κ and some of their properties (in Russian, English summary), Zapiski Nauchnykh Seminarov Leningradskogo Otdelinya Matematicheskogo Instituta im. V.A. Steklova AN SSSR, 130, 175-180 (1983) · Zbl 0529.60010 |
| [9] | Philippou, A. N., The negative binomial distribution of order κ and some of its properties, Biometrical Journal, 26, 789-794 (1984) · Zbl 0566.60014 |
| [10] | Philippou, A. N., On multiparameter distributions of order κ, Annals of the Institute of Statistical Mathematics (1987), (accepted) |
| [11] | Philippou, A. N.; Georghiou, C.; Philippou, G. N., A generalized geometric distribution and some of its properties, Statistics & Probability Letters, 1, 171-175 (1983) · Zbl 0517.60010 |
| [12] | Sibuya, M.; Yoshimura, I.; Shimizu, R., Negative multinomial distribution, Annals of the Institute of Statistical Mathematics, 16, 409-426 (1964) · Zbl 0146.39303 |
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