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Multivariate discrete distributions via sums and shares

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  • Jones, M.C.
  • Marchand, Éric

Abstract

In this article, we develop a sum and share decomposition to model multivariate discrete distributions, and more specifically multivariate count data that can be divided into a number of distinct categories. From a Poisson mixture model for the sum and a multinomial mixture model for the shares, a rich ensemble of properties, examples and relationships arises. As a main example, a seemingly new multivariate model involving a negative binomial sum and Pólya shares is considered, previously seen only in the bivariate case, for which we present two contrasting applications. For other choices of the distribution of the sum, natural but novel discrete multivariate Liouville distributions emerge; an important special case of these is that of Schur constant distributions. Analogies and interactions with related continuous distributions are to the fore throughout.

Suggested Citation

  • Jones, M.C. & Marchand, Éric, 2019. "Multivariate discrete distributions via sums and shares," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 83-93.
    • Handle: RePEc:eee:jmvana:v:171:y:2019:i:c:p:83-93
    DOI: 10.1016/j.jmva.2018.11.011
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    References listed on IDEAS

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    1. Sellers, Kimberly F. & Morris, Darcy Steeg & Balakrishnan, Narayanaswamy, 2016. "Bivariate Conway–Maxwell–Poisson distribution: Formulation, properties, and inference," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 152-168.
    2. Castañer, A. & Claramunt, M.M. & Lefèvre, C. & Loisel, S., 2015. "Discrete Schur-constant models," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 343-362.
    3. Felix Famoye & P. Consul, 1995. "Bivariate generalized Poisson distribution with some applications," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 42(1), pages 127-138, December.
    4. McNeil, Alexander J. & Neslehová, Johanna, 2010. "From Archimedean to Liouville copulas," Journal of Multivariate Analysis, Elsevier, vol. 101(8), pages 1772-1790, September.
    5. Aoudia, Djilali Ait & Marchand, Éric & Perron, François, 2016. "Counts of Bernoulli success strings in a multivariate framework," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 1-10.
    6. Gupta, Rameshwar D. & Richards, Donald St.P., 1987. "Multivariate Liouville distributions," Journal of Multivariate Analysis, Elsevier, vol. 23(2), pages 233-256, December.
    7. Djilali Ait Aoudia & Éric Marchand, 2014. "On a Simple Construction of a Bivariate Probability Function With a Common Marginal," The American Statistician, Taylor & Francis Journals, vol. 68(3), pages 170-173, February.
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    Cited by:

    1. Stoltenberg, Emil Aas & Hjort, Nils Lid, 2020. "Multivariate estimation of Poisson parameters," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    2. Claude Lefèvre & Matthieu Simon, 2021. "Schur-Constant and Related Dependence Models, with Application to Ruin Probabilities," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 317-339, March.
    3. Peyhardi, Jean & Fernique, Pierre & Durand, Jean-Baptiste, 2021. "Splitting models for multivariate count data," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
    4. Castañer, Anna & Claramunt, M. Mercè & Lefèvre, Claude & Loisel, Stéphane, 2019. "Partially Schur-constant models," Journal of Multivariate Analysis, Elsevier, vol. 172(C), pages 47-58.

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