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Matrix-Form Recursions for a Family of Compound Distributions

Published online by Cambridge University Press:  09 August 2013

Xueyuan Wu  and
Shuanming Li
Xueyuan Wu
Affiliation:
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Victoria 3010, Australia, E-mail: shli@unimelb.edu.au, Fax: 61-3-83446899
Shuanming Li
Affiliation:
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Victoria 3010, Australia, E-mail: shli@unimelb.edu.au, Fax: 61-3-83446899
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Abstract

In this paper, we aim to evaluate the distribution of the aggregate claims in the collective risk model. The claim count distribution is firstly assumed to belong to a generalised (a, b, 0) family. A matrix form recursive formula is then derived to evaluate the related compound distribution when individual claim amounts follow a discrete distribution on non-negative integers. The corresponding formula is also given for continuous individual claim amounts. Secondly, we pay particular attention to the recursive formula for compound phase-type distributions, since only certain types of discrete phase-type distributions belong to the generalised (a, b, 0) family. Similar recursive formulae are obtained for discrete and continuous individual claim amount distributions. Finally, numerical examples are presented for three counting distributions.

Keywords

Discrete phase-type distributionsGeneralised (a, b, 0) familyRecursive formulaCompound distribution
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 40 , Issue 1 , May 2010 , pp. 351 - 368
Copyright
Copyright © International Actuarial Association 2010

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