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On a Multiplicative Multivariate Gamma Distribution with Applications in Insurance

Author

Listed:
  • Vadim Semenikhine

    (Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada)

  • Edward Furman

    (Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada)

  • Jianxi Su

    (Department of Statistics, Purdue University, West Lafayette, IN 47906, USA)

Abstract

One way to formulate a multivariate probability distribution with dependent univariate margins distributed gamma is by using the closure under convolutions property. This direction yields an additive background risk model, and it has been very well-studied. An alternative way to accomplish the same task is via an application of the Bernstein–Widder theorem with respect to a shifted inverse Beta probability density function. This way, which leads to an arguably equally popular multiplicative background risk model (MBRM), has been by far less investigated. In this paper, we reintroduce the multiplicative multivariate gamma (MMG) distribution in the most general form, and we explore its various properties thoroughly. Specifically, we study the links to the MBRM, employ the machinery of divided differences to derive the distribution of the aggregate risk random variable explicitly, look into the corresponding copula function and the measures of nonlinear correlation associated with it, and, last but not least, determine the measures of maximal tail dependence. Our main message is that the MMG distribution is (1) very intuitive and easy to communicate, (2) remarkably tractable, and (3) possesses rich dependence and tail dependence characteristics. Hence, the MMG distribution should be given serious considerations when modelling dependent risks.

Suggested Citation

  • Vadim Semenikhine & Edward Furman & Jianxi Su, 2018. "On a Multiplicative Multivariate Gamma Distribution with Applications in Insurance," Risks, MDPI, vol. 6(3), pages 1-20, August.
    • Handle: RePEc:gam:jrisks:v:6:y:2018:i:3:p:79-:d:163347
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    References listed on IDEAS

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    Cited by:

    1. 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.
    2. Nadezhda Gribkova & Ričardas Zitikis, 2019. "Statistical detection and classification of background risks affecting inputs and outputs," METRON, Springer;Sapienza Università di Roma, vol. 77(1), pages 1-18, April.
    3. Nadezhda Gribkova & Ričardas Zitikis, 2018. "A User-Friendly Algorithm for Detecting the Influence of Background Risks on a Model," Risks, MDPI, vol. 6(3), pages 1-11, September.

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