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Risks 2018, 6(3), 79; https://doi.org/10.3390/risks6030079

On a Multiplicative Multivariate Gamma Distribution with Applications in Insurance

1
Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada
2
Department of Statistics, Purdue University, West Lafayette, IN 47906, USA
*
Author to whom correspondence should be addressed.
Received: 13 July 2018 / Revised: 8 August 2018 / Accepted: 8 August 2018 / Published: 12 August 2018
(This article belongs to the Special Issue Risk, Ruin and Survival: Decision Making in Insurance and Finance)
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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. View Full-Text
Keywords: multivariate gamma distribution; multiplicative background risk model; aggregate risk; individual risk model; collective risk model multivariate gamma distribution; multiplicative background risk model; aggregate risk; individual risk model; collective risk model
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Semenikhine, V.; Furman, E.; Su, J. On a Multiplicative Multivariate Gamma Distribution with Applications in Insurance. Risks 2018, 6, 79.

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