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Modeling discrete common-shock risks through matrix distributions

Published online by Cambridge University Press:  16 September 2025

Martin Bladt ,
Eric C. K. Cheung ,
Oscar Peralta Open the ORCID record for Oscar Peralta [Opens in a new window]  and
Jae-Kyung Woo
Martin Bladt
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark
Eric C. K. Cheung
Affiliation:
School of Risk and Actuarial Studies, UNSW Business School, University of New South Wales, Sydney, Australia
Oscar Peralta*
Affiliation:
Department of Actuarial and Insurance Sciences, Autonomous Technological Institute of México, México City, México
Jae-Kyung Woo
Affiliation:
School of Risk and Actuarial Studies, UNSW Business School, University of New South Wales, Sydney, Australia
*
Corresponding author: Oscar Peralta; Email: oscar.peralta@itam.mx
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Abstract

We introduce a novel class of bivariate common-shock discrete phase-type (CDPH) distributions to describe dependencies in loss modeling, with an emphasis on those induced by common shocks. By constructing two jointly evolving terminating Markov chains that share a common evolution up to a random time corresponding to the common shock component, and then proceed independently, we capture the essential features of risk events influenced by shared and individual-specific factors. We derive explicit expressions for the joint distribution of the termination times and prove various class and distributional properties, facilitating tractable analysis of the risks. Extending this framework, we model random sums where aggregate claims are sums of continuous phase-type random variables with counts determined by these termination times and show that their joint distribution belongs to the multivariate phase-type or matrix-exponential class. We develop estimation procedures for the CDPH distributions using the expectation-maximization algorithm and demonstrate the applicability of our models through simulation studies and an application to bivariate insurance claim frequency data. In particular, the distribution of the latent common shock component present in correlated count data can be estimated as well.

Keywords

Phase-type distributiondiscrete bivariate distributioncommon shocksexpectation-maximization algorithm

Information

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , First View , pp. 1 - 26
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The International Actuarial Association

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