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Multivariate Exponential and Geometric Distributions with Limited Memory


  • Marshall, A. W.
  • Olkin, I.


Many connections between geometric and exponential distributions are known. Characterizations and derivations of these distributions often run parallel. Moreover, one kind of distribution can be derived from the other: exponential distributions are limits of sequences of rescaled geometric distributions, and the integral parts of exponential random variables have geometric distributions. The lack of memory property is expressed by functional equations of the form (x + u) = (x) (u) for all (x, u) [set membership, variant] , where (x) = P{X1 > x1, ..., Xn > xn}. With = 2n+, the equation expresses a complete lack of memory that is possessed only by distributions with independent exponential marginals. But when is a proper subset of 2n+, the functional equation expresses a partial lack of memory property that in some cases is possessed by a more interesting family of multivariate exponential distributions, as for example, those with exponential minima. In this paper appropriate choices for and the resulting families of solutions are investigated, together with the associated families of multivariate geometric distributions.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:53:y:1995:i:1:p:110-125

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    References listed on IDEAS

    1. Dielman, Terry E. & Rose, Elizabeth L., 1995. "A bootstrap approach to hypothesis testing in least absolute value regression," Computational Statistics & Data Analysis, Elsevier, vol. 20(2), pages 119-130, August.
    2. Dielman, Terry E. & Rose, Elizabeth L., 1996. "A note on hypothesis testing in LAV multiple regression: A small sample comparison," Computational Statistics & Data Analysis, Elsevier, vol. 21(4), pages 463-470, April.
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    Cited by:

    1. Koutras, M.V. & Maravelakis, P.E. & Bersimis, S., 2008. "Techniques for controlling bivariate grouped observations," Journal of Multivariate Analysis, Elsevier, vol. 99(7), pages 1474-1488, August.
    2. Shenkman, Natalia, 2017. "A natural parametrization of multivariate distributions with limited memory," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 234-251.
    3. Brigo, Damiano & Mai, Jan-Frederik & Scherer, Matthias, 2016. "Markov multi-variate survival indicators for default simulation as a new characterization of the Marshall–Olkin law," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 60-66.
    4. Cossette, Hélène & Marceau, Etienne & Perreault, Samuel, 2015. "On two families of bivariate distributions with exponential marginals: Aggregation and capital allocation," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 214-224.
    5. Marceau, Etienne, 2009. "On the discrete-time compound renewal risk model with dependence," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 245-259, April.
    6. 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.

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