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COPULA REPRESENTATIONS FOR THE SUM OF DEPENDENT RISKS: MODELS AND COMPARISONS

Published online by Cambridge University Press:  23 December 2020

Jorge Navarro Open the ORCID record for Jorge Navarro [Opens in a new window]  and
José María Sarabia Open the ORCID record for José María Sarabia [Opens in a new window]
Jorge Navarro
Affiliation:
Department of Statistics and Operational Research, University of Murcia, 30100Murcia, Spain E-mail: jorgenav@um.es
José María Sarabia
Affiliation:
Department of Quantitative Methods, CUNEF University, 28040Madrid, Spain E-mail: josemaria.sarabia@cunef.edu
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Abstract

The study of the distributions of sums of dependent risks is a key topic in actuarial sciences, risk management, reliability and in many branches of applied and theoretical probability. However, there are few results where the distribution of the sum of dependent random variables is available in a closed form. In this paper, we obtain several analytical expressions for the distribution of the aggregated risks under dependence in terms of copulas. We provide several representations based on the underlying copula and the marginal distribution functions under general hypotheses and in any dimension. Then, we study stochastic comparisons between sums of dependent risks. Finally, we illustrate our theoretical results by studying some specific models obtained from Clayton, Ali-Mikhail-Haq and Farlie-Gumbel-Morgenstern copulas. Extensions to more general copulas are also included. Bounds and the limiting behavior of the hazard rate function for the aggregated distribution of some copulas are studied as well.

Keywords

convolutionC-convolutiondistorted distributionshazard ratestochastic orders
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 2 , April 2022 , pp. 320 - 340
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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