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Some properties and characterizations for generalized multivariate Pareto distributions

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  • Yeh, Hsiaw-Chan

Abstract

In this paper, several distributional properties and characterization theorems of the generalized multivariate Pareto distributions are studied. It is found that the multivariate Pareto distributions have many mixture properties. They are mixed either by geometric, Weibull, or exponential variables. The multivariate Pareto, MP(k)(I), MP(k)(II), and MP(k)(IV) families have closure property under finite sample minima. The MP(k)(III) family is closed under both geometric minima and geometric maxima. Through the geometric minima procedure, one characterization theorem for MP(k)(III) distribution is developed. Moreover, the MP(k)(III) distribution is proved as the limit multivariate distribution under repeated geometric minimization. Also, a characterization theorem for the homogeneous MP(k)(IV) distribution via the weighted minima among the ordered coordinates is developed. Finally, the MP(k)(II) family is shown to have the truncation invariant property.

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  • Yeh, Hsiaw-Chan, 2004. "Some properties and characterizations for generalized multivariate Pareto distributions," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 47-60, January.
    • Handle: RePEc:eee:jmvana:v:88:y:2004:i:1:p:47-60
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    References listed on IDEAS

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    1. Yeh, Hsiaw-Chan, 2002. "Six multivariate Zipf distributions and their related properties," Statistics & Probability Letters, Elsevier, vol. 56(2), pages 131-141, January.
    2. Yeh, H. C., 1994. "Some Properties of the Homogeneous Multivariate Pareto (IV) Distribution," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 46-53, October.
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    Cited by:

    1. Alexandru V. Asimit & Raluca Vernic & Riċardas Zitikis, 2013. "Evaluating Risk Measures and Capital Allocations Based on Multi-Losses Driven by a Heavy-Tailed Background Risk: The Multivariate Pareto-II Model," Risks, MDPI, vol. 1(1), pages 1-20, March.
    2. Yeh, Hsiaw-Chan, 2009. "Multivariate semi-Weibull distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1634-1644, September.
    3. Yeh, Hsiaw-Chan, 2010. "Multivariate semi-logistic distributions," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 893-908, April.
    4. Yeh, Hsiaw-Chan, 2007. "Three general multivariate semi-Pareto distributions and their characterizations," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1305-1319, July.
    5. Gribkova, N.V. & Su, J. & Zitikis, R., 2022. "Inference for the tail conditional allocation: Large sample properties, insurance risk assessment, and compound sums of concomitants," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 199-222.
    6. Arendarczyk, Marek & Kozubowski, Tomasz. J. & Panorska, Anna K., 2018. "The joint distribution of the sum and maximum of dependent Pareto risks," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 136-156.
    7. Raluca Vernic, 2011. "Tail Conditional Expectation for the Multivariate Pareto Distribution of the Second Kind: Another Approach," Methodology and Computing in Applied Probability, Springer, vol. 13(1), pages 121-137, March.

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