On a Multiplicative Multivariate Gamma Distribution with Applications in Insurance

Abstract

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.

Special Issue “Risk, Ruin and Survival: Decision Making in Insurance and Finance”

“…In particular, practitioners often wish to work with multivariate distributions that (i) admit meaningful and relevant interpretations, (ii) allow for adequate fits (marginally or jointly) to a wide range of multivariate data, (iii) possess desirable distributional properties for insurance valuation and risk management, and (iv) can be readily implemented. The multivariate gamma family proposed by Semenikhine et al (2018) is exactly such. Although the family is exceptionally general, the authors have succeeded in thoroughly exploring its properties.…”

“…Although the family is exceptionally general, the authors have succeeded in thoroughly exploring its properties. In particular, Semenikhine et al (2018) have linked the family to the multiplicative background risk model, derived an explicit formula for the distribution of the aggregate risk, specified the corresponding copula function, and determined measures of nonlinear correlation, including the index of maximal tail dependence (Furman et al 2015).…”

Risk, Ruin and Survival

“…Although the family is exceptionally general, the authors have succeeded in thoroughly exploring its properties. In particular, Semenikhine et al (2018) have linked the family to the multiplicative background risk model, derived an explicit formula for the distribution of the aggregate risk, specified the corresponding copula function, and determined measures of nonlinear correlation, including the index of maximal tail dependence .…”

“…Various models have been proposed, including additive, multiplicative, and more intricate ones that couple underlying losses (or, generally speaking, inputs) with background risks. For recent far-reaching contributions to this area, we refer to Perote et al (2015), Su (2016), Su and Furman (2017a, 2017b) Semenikhine et al (2018), , as well as to the extensive lists of references therein.…”

A User-Friendly Algorithm for Detecting the Influence of Background Risks on a Model

“…Various models have been proposed, including additive, multiplicative, and more intricate ones that couple underlying losses (or, generally speaking, inputs) with background risks. For recent far-reaching contributions to this area, we refer to Perote et al (2015), Su (2016), Su andFurman (2017a, 2017b) Semenikhine et al (2018), , as well as to the extensive lists of references therein.…”