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Matrix-form Recursive Evaluation of the Aggregate Claims Distribution Revisited

Published online by Cambridge University Press:  20 April 2011

Kok Keng Siaw ,
Xueyuan Wu ,
David Pitt  and
Yan Wang
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Abstract

This paper aims to evaluate the aggregate claims distribution under the collective risk model when the number of claims follows a so-called generalised (a, b, 1) family distribution. The definition of the generalised (a, b, 1) family of distributions is given first, then a simple matrix-form recursion for the compound generalised (a, b, 1) distributions is derived to calculate the aggregate claims distribution with discrete non-negative individual claims. Continuous individual claims are discussed as well and an integral equation of the aggregate claims distribution is developed. Moreover, a recursive formula for calculating the moments of aggregate claims is also obtained in this paper. With the recursive calculation framework being established, members that belong to the generalised (a, b, 1) family are discussed. As an illustration of potential applications of the proposed generalised (a, b, 1) distribution family on modelling insurance claim numbers, two numerical examples are given. The first example illustrates the calculation of the aggregate claims distribution using a matrix-form Poisson for claim frequency with logarithmic claim sizes. The second example is based on real data and illustrates maximum likelihood estimation for a set of distributions in the generalised (a, b, 1) family.

Keywords

(a, b, 1) FamilyGeneralised (a, b, 1) FamilyRecursive FormulaCompound Distributions
Type
Papers
Information
Annals of Actuarial Science , Volume 5 , Issue 2 , September 2011 , pp. 163 - 179
Copyright
Copyright © Institute and Faculty of Actuaries 2011

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References

Bermúdez, L., Karlis, D. (2011). Bayesian multivariate Poisson models for insurance ratemaking. Insurance: Mathematics and Economics, 48, 226–236.Google Scholar
Bermúdez, L. (2009). A priori ratemaking using bivariate Poisson regression models. Insurance: Mathematics and Economics, 44, 135–141.Google Scholar
Boucher, J.P., Denuit, M., Guillén, M. (2009). Number of accidents or number of claims? An approach with zero-inflated Poisson models for panel data. Journal of Risk and Insruance, 76(4), 821–846.CrossRefGoogle Scholar
Brouhns, N., Denuit, M., Guillén, M., Pinquet, J. (2003). Bonus-malus scales in segmented tariffs with stochastic migration between segments. Journal of Risk and Insurance, 70(4), 577–599.CrossRefGoogle Scholar
Culver, W.J. (1966). On the existence and uniqueness of the real logarithm of a matrix. Proceedings of the American Mathematical Society, 17, 1146–1151.CrossRefGoogle Scholar
Klugman, S.A., Panjer, H.H., Willmot, G.E. (1998). Loss Models: From Data to Decisions. Wiley Series in Probability and Statistics, John Wiley and Sons, Inc.Google Scholar
Latouche, G., Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling. ASA SIAM, Philadelphia.CrossRefGoogle Scholar
Neuts, M.F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Johns Hopkins University Press, Baltimore.Google Scholar
Panjer, H. (1981). Recursive evaluation of a family of compound distributions. ASTIN Bulletin, 12, 22–26.CrossRefGoogle Scholar
Pinquet, J., Guillén, M., Bolancé, C. (2001). Long-range contagion in automobile insurance data: estimation and implications for experience rating. ASTIN Bulletin, 31(2), 337–348.CrossRefGoogle Scholar
Schröter, K.J. (1990). On a family of counting distributions and recursions for the related compound distributions. Scandinavian Actuarial Journal, 161–175.CrossRefGoogle Scholar
Sundt, B. (2002). Recursive evaluation of aggregate claims distributions. Insurance: Mathematics and Economics, 30, 297–332.Google Scholar
Sundt, B. (2003). Some recursions for moments of compound distributions. Insurance: Mathematics and Economics, 33, 486–496.Google Scholar
Sundt, B., Jewell, W.S. (1981). Further results on recursive evaluation of compound distributions. ASTIN Bulletin, 12, 27–39.CrossRefGoogle Scholar
Sundt, B., Vernic, R. (2009). Recursions for convolutions and compound distributions with insurance applications. Springer-Verlag, Heidelberg.Google Scholar
Willmot, G. (1988). Sundt and Jewell's family of discrete distributions. ASTIN Bulletin, 18, 17–29.CrossRefGoogle Scholar
Wu, X., Li, S. (2010). Matrix-form recursions for a family of compound distributions. ASTIN Bulletin, 40, 351–368.CrossRefGoogle Scholar