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On a multivariate Pareto distribution

Author

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  • Asimit, Alexandru V.
  • Furman, Edward
  • Vernic, Raluca

Abstract

A multivariate distribution possessing arbitrarily parameterized Pareto margins is formulated and studied. The distribution is believed to allow for an adequate modeling of dependent heavy tailed risks with a non-zero probability of simultaneous loss. Numerous links to certain existing probabilistic models, as well as seemingly useful characteristic results are proved. Expressions for, e.g., decumulative distribution functions, densities, (joint) moments and regressions are developed. An application to the classical pricing problem is considered, and some formulas are derived using the recently introduced economic weighted premium calculation principles.

Suggested Citation

  • Asimit, Alexandru V. & Furman, Edward & Vernic, Raluca, 2010. "On a multivariate Pareto distribution," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 308-316, April.
    • Handle: RePEc:eee:insuma:v:46:y:2010:i:2:p:308-316
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    References listed on IDEAS

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    1. Peter Xue‐Kun Song, 2000. "Multivariate Dispersion Models Generated From Gaussian Copula," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(2), pages 305-320, June.
    2. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted premium calculation principles," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 459-465, February.
    3. Genest, Christian & Nešlehová, Johanna, 2007. "A Primer on Copulas for Count Data," ASTIN Bulletin, Cambridge University Press, vol. 37(2), pages 475-515, November.
    4. Pfeifer, Dietmar & Nešlehová, Johana, 2004. "Modeling and Generating Dependent Risk Processes for IRM and DFA," ASTIN Bulletin, Cambridge University Press, vol. 34(2), pages 333-360, November.
    5. Vernic, Raluca, 1997. "On The Bivariate Generalized Poisson Distribution," ASTIN Bulletin, Cambridge University Press, vol. 27(1), pages 23-32, May.
    6. Tsanakas, Andreas, 2008. "Risk measurement in the presence of background risk," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 520-528, April.
    7. Cai, Jun & Tan, Ken Seng, 2007. "Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures," ASTIN Bulletin, Cambridge University Press, vol. 37(1), pages 93-112, May.
    8. Furman, Edward & Landsman, Zinoviy, 2008. "Economic Capital Allocations for Non-negative Portfolios of Dependent Risks," ASTIN Bulletin, Cambridge University Press, vol. 38(2), pages 601-619, November.
    9. Bäuerle, Nicole & Grübel, Rudolf, 2005. "Multivariate Counting Processes: Copulas and Beyond," ASTIN Bulletin, Cambridge University Press, vol. 35(2), pages 379-408, November.
    10. Laeven, Roger J.A. & Goovaerts, Marc J. & Hoedemakers, Tom, 2005. "Some asymptotic results for sums of dependent random variables, with actuarial applications," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 154-172, October.
    11. Vernic, Raluca, 2000. "A Multivariate Generalization of the Generalized Poisson Distribution," ASTIN Bulletin, Cambridge University Press, vol. 30(1), pages 57-67, May.
    12. Edward Frees & Emiliano Valdez, 1998. "Understanding Relationships Using Copulas," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 1-25.
    13. Furman, Edward, 2008. "On a multivariate gamma distribution," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2353-2360, October.
    14. de Lourdes Centeno, Maria, 2005. "Dependent risks and excess of loss reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 229-238, October.
    15. Chiragiev, Arthur & Landsman, Zinoviy, 2009. "Multivariate flexible Pareto model: Dependency structure, properties and characterizations," Statistics & Probability Letters, Elsevier, vol. 79(16), pages 1733-1743, August.
    16. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted risk capital allocations," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 263-269, October.
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    Cited by:

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    3. Yang, Xipei & Frees, Edward W. & Zhang, Zhengjun, 2011. "A generalized beta copula with applications in modeling multivariate long-tailed data," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 265-284, September.
    4. Sarabia, José María & Gómez-Déniz, Emilio & Prieto, Faustino & Jordá, Vanesa, 2016. "Risk aggregation in multivariate dependent Pareto distributions," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 154-163.
    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. Su, Jianxi & Furman, Edward, 2017. "Multiple risk factor dependence structures: Distributional properties," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 56-68.
    7. 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.
    8. Furman, Edward & Kye, Yisub & Su, Jianxi, 2021. "Multiplicative background risk models: Setting a course for the idiosyncratic risk factors distributed phase-type," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 153-167.
    9. Nadarajah, Saralees & Afuecheta, Emmanuel & Chan, Stephen, 2013. "A double generalized Pareto distribution," Statistics & Probability Letters, Elsevier, vol. 83(12), pages 2656-2663.
    10. Woo, Jae-Kyung, 2016. "On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 354-363.
    11. Cheung, Eric C.K. & Peralta, Oscar & Woo, Jae-Kyung, 2022. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 364-389.
    12. Su, Jianxi & Furman, Edward, 2017. "Multiple risk factor dependence structures: Copulas and related properties," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 109-121.

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