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Multivariate geometric distributions. (English) Zbl 0900.62271
Summary: Families of multivariate geometric distributions with flexible correlations can be constructed by applying inverse sampling to a sequence of multinomial trials, and counting outcomes in possibly overlapping categories. Further multivariate families can be obtained by considering other stopping rules, with the possibility of different stopping rules for different counts. A simple characterisation is given for stopping rules which produce joint distributions with marginals having the same form as that of the number of trials. The inverse sampling approach provides a unified treatment of diverse results presented by earlier authors, including Guldberg (1934), Bates and Neyman (1952), Edwards and Gurland (1961), Hawkes (1972), Paulson and Uppuluri (1972) and Griffiths and Milne (1987). It also provides a basis for investigating the range of possible correlations for a given set of marginal parameters. In the case of more than two joint geometric or negative binomial variables, a convenient matrix formulation is provided.

MSC:
62H10 Multivariate distribution of statistics
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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References:
[1] Bates G.E., Univ. of California Publications in Statistics 1 pp 255– (1952)
[2] Block H.W., The Theory and Applications of Reliability 1 (1977)
[3] Doss D.C., J. Multivariate Anal 9 pp 460– (1979)
[4] Griffiths R.C., J. Mull, Anal 22 pp 13– (1987)
[5] Goldberg A., Skand Aktuarietidskr 17 pp 89– (1934)
[6] Hawkes A.G., J. Royal Statist. Soc.t Ser. B 34 pp 129– (1972)
[7] Johnson N.L., Urn Models and Their Application (1977)
[8] Marshall A.W., J. Amer. Statist, Ass 80 pp 332– (1985)
[9] Patil G.P., A Dictionary and Bibliography of Discrete Distributions (1968) · Zbl 0193.18301
[10] Paulson A.S., Sankhya, Ser. A 34 pp 88– (1972)
[11] Sibuya M., Annals Institute of Statist, Math 16 pp 409– (1964)
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