×

zbMATH — the first resource for mathematics

Multivariate analogs of classical univariate discrete distributions and their properties. (English) Zbl 1419.60015
Summary: Some discrete distributions such as Bernoulli, binomial, geometric, negative binomial, Poisson, Polya-Aeppli, and others play an important role in applied problems of probability theory and mathematical statistics. We propose a variant of a multivariate distribution whose components have a given univariate discrete distribution. In fact we consider some very general variant of the so-called reduction method. We find the explicit form of the mass function and generating function of such distribution and study their properties. We prove that our construction is unique in natural exponential families of distributions. Our results are the generalization and unification of many results of other authors.
MSC:
60E05 Probability distributions: general theory
62H10 Multivariate distribution of statistics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Campbell, JT, Poisson correlation function, Proc. Edinburg Math. Soc., 4, 18-26, (1934) · JFM 60.0466.01
[2] Ivanova, NL, The reconstruction of natural exponential families by its marginals, J. Math. Sci., 106, 2672-2681, (2001) · Zbl 0995.60022
[3] Ivanova, NL; Khokhlov, YS, Reconstruction of multivariate distribution with Poisson components, Moscow Univ. Comput. Math. Cybern., 25, 37-42, (2001) · Zbl 0996.60004
[4] Ivanova, NL; Khokhlov, YS, The new variant of multivariate generalization of the generalized Poisson distribution, J. Inform. Proc., 2, 193-194, (2002)
[5] Ivanova, NL; Khokhlov, YS, Multidimensional collective risk model, J. Math. Sci., 146, 6000-6007, (2007)
[6] N. L. Johnson, S. Kotz, and N. Balakrishnan, Discrete Multivariate Distributions, Wiley, New York (1997). · Zbl 0868.62048
[7] S. Kocherlakota and K. Kocherlakota, Bivariate Discrete Distributions, Marcel Dekker, New York (1992). · Zbl 0794.62002
[8] Krummenauer, F., Limit theorems for multivariate discrete distributions, Metrika, 47, 47-69, (1998) · Zbl 1092.60504
[9] Krummenauer, F., Representation of multivariate discrete distributions by probability generating functions, Stat. Probab. Lett., 39, 327-331, (1998) · Zbl 0916.62041
[10] Marshall, AW; Olkin, I., A family of bivariate distributions generated by the Bernoulli distribution, J. Am. Stat. Assoc., 80, 332-338, (1985) · Zbl 0575.60023
[11] Minkova, LD; Balakrishnan, N., On a bivariate Polya-Aeppli distribution, Comput. Stat. Theor. Methods, 43, 5026-5038, (2014) · Zbl 1307.60006
[12] Minkova, LD; Balakrishnan, N., Type II bivariate Polya-Aeppli distribution, Stat. Probab. Lett., 88, 40-49, (2014) · Zbl 1294.60021
[13] Ong, SH; Lee, PH, The non-central negative binomial distribution, Biometrical J., 21, 611-628, (1979) · Zbl 0432.62011
[14] Panjer, HH, Recursive evaluation of a family of compound distributions, ASTIN Bull., 12, 22-26, (1981)
[15] Vernic, R., A multivariate generalization of the generalized Poisson distribution, ASTIN Bull., 30, 57-67, (2000) · Zbl 1114.62332
[16] Davy, PJ; Ranger, JCW, Multivariate geometric distributions, Comput. Stat. Theor. Methods, 25, 2971-2987, (1996) · Zbl 0900.62271
[17] Zolotukhin, IV, Discrete analogue of Marshall-Olkin multivariate exponential distribution, Obozrenie Prikladnoj i Promyshlennoj Matematiki, 23, 150-151, (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.