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COMPARISON OF APPROXIMATIONS FOR COMPOUND POISSON PROCESSES

Published online by Cambridge University Press:  25 June 2015

Raffaello Seri  and
Christine Choirat
Raffaello Seri*
Affiliation:
Dipartimento di Economia, Università degli Studi dell'Insubria, Via Monte Generoso 71, 21100 Varese, Italy
Christine Choirat
Affiliation:
IQSS Harvard University, 1737 Cambridge Street, CGIS Knafel Building, Cambridge, MA 02138, USA E-Mail: [email protected]
*
E-Mail: [email protected]
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Abstract

In this paper, we compare the error in several approximation methods for the cumulative aggregate claim distribution customarily used in the collective model of insurance theory. In this model, it is usually supposed that a portfolio is at risk for a time period of length t. The occurrences of the claims are governed by a Poisson process of intensity μ so that the number of claims in [0,t] is a Poisson random variable with parameter λ = μ t. Each single claim is an independent replication of the random variable X, representing the claim severity. The aggregate claim or total claim amount process in [0,t] is represented by the random sum of N independent replications of X, whose cumulative distribution function (cdf) is the object of study. Due to its computational complexity, several approximation methods for this cdf have been proposed. In this paper, we consider 15 approximations put forward in the literature that only use information on the lower order moments of the involved distributions. For each approximation, we consider the difference between the true distribution and the approximating one and we propose to use expansions of this difference related to Edgeworth series to measure their accuracy as λ = μ t diverges to infinity. Using these expansions, several statements concerning the quality of approximations for the distribution of the aggregate claim process can find theoretical support. Other statements can be disproved on the same grounds. Finally, we investigate numerically the accuracy of the proposed formulas.

Keywords

Collective risk theorycompound Poisson processEdgeworth series
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 3 , September 2015 , pp. 601 - 637
Copyright
Copyright © Astin Bulletin 2015 

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