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AGGREGATION OF DEPENDENT RISKS IN MIXTURES OF EXPONENTIAL DISTRIBUTIONS AND EXTENSIONS

Published online by Cambridge University Press:  25 April 2018

José María Sarabia ,
Emilio Gómez-Déniz ,
Faustino Prieto  and
Vanesa Jordá
José María Sarabia*
Affiliation:
Department of Economics, University of Cantabria, Avda de los Castros s/n, 39005 Santander, Spain
Emilio Gómez-Déniz
Affiliation:
Department of Quantitative Methods in Economics and TiDES Institute, University of Las Palmas de Gran Canaria, 35017 Las Palmas de G.C., Spain E-Mail: [email protected]
Faustino Prieto
Affiliation:
Department of Economics, University of Cantabria, Avda de los Castros s/n, 39005 Santander, Spain E-Mail: [email protected]
Vanesa Jordá
Affiliation:
Department of Economics, University of Cantabria, Avda de los Castros s/n, 39005 Santander, Spain E-Mail: [email protected]
*
E-Mail: [email protected]
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Abstract

The distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and in many branches of applied probability. In this paper, we obtain analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modelled according to a mixture of exponential distributions. We first review the properties of the multivariate mixture of exponential distributions, to then obtain the analytical formulation for the pdf and the cdf for the aggregated distribution. We study in detail some specific families with Pareto (Sarabia et al., 2016), gamma, Weibull and inverse Gaussian mixture of exponentials (Whitmore and Lee, 1991) claims. We also discuss briefly the computation of risk measures, formulas for the ruin probability (Albrecher et al., 2011) and the collective risk model. An extension of the basic model based on mixtures of gamma distributions is proposed, which is one of the suggested directions for future research.

Keywords

Aggregationdependent random variablesLaplace transformcollective risk modelpartial Bell polynomials
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 3 , September 2018 , pp. 1079 - 1107
Copyright
Copyright © Astin Bulletin 2018 

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