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A natural parametrization of multivariate distributions with limited memory

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  • Shenkman, Natalia

Abstract

A unified formulation of the theory of d-variate wide-sense geometric (GdW) and Marshall–Olkin exponential (MOd) distributions is presented in which d-monotone set functions occupy a central role. A semi-analytical derivation of GdW and MOd distributions is deduced directly from the lack-of-memory property. In this context, the distributions are parametrized with d-monotone and d-log-monotone set functions arising from the univariate marginal distributions of minima and the d-decreasingness of the survival functions. In addition, a one-to-one correspondence is established between d-monotone (resp. d-log-monotone) set functions and d-variate (resp. d-variate min-infinitely divisible) Bernoulli distributions. The advantage of such a parametrization is that it makes the distributions highly tractable. As a showcase, we derive new results on the minimum stability and divisibility of the GdW family, and on the marginal equivalence in minima of GdW and distributions with geometric minima. Similarly, a surprisingly simple proof is given of the prominent result of Esary and Marshall (1974) on the marginal equivalence in minima of multivariate exponential distributions.

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  • Shenkman, Natalia, 2017. "A natural parametrization of multivariate distributions with limited memory," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 234-251.
    • Handle: RePEc:eee:jmvana:v:155:y:2017:i:c:p:234-251
    DOI: 10.1016/j.jmva.2017.01.004
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    References listed on IDEAS

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    1. Marshall, Albert W. & Olkin, Ingram, 1991. "Functional equations for multivariate exponential distributions," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 209-215, October.
    2. Ressel, Paul, 2011. "Monotonicity properties of multivariate distribution and survival functions -- With an application to Lévy-frailty copulas," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 393-404, March.
    3. Mai, Jan-Frederik & Scherer, Matthias, 2009. "Lévy-frailty copulas," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1567-1585, August.
    4. Marshall, A. W. & Olkin, I., 1995. "Multivariate Exponential and Geometric Distributions with Limited Memory," Journal of Multivariate Analysis, Elsevier, vol. 53(1), pages 110-125, April.
    5. Mai, Jan-Frederik & Scherer, Matthias & Shenkman, Natalia, 2013. "Multivariate geometric distributions, (logarithmically) monotone sequences, and infinitely divisible laws," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 457-480.
    6. Colangelo, Antonio & Scarsini, Marco & Shaked, Moshe, 2005. "Some notions of multivariate positive dependence," Insurance: Mathematics and Economics, Elsevier, vol. 37(1), pages 13-26, August.
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