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Matrix-form Recursive Evaluation of the Aggregate Claims Distribution Revisited

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

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.

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Corresponding author

Contact address Kok Keng Siaw, School of Banking and Finance, Australian School of Business, The University of New South Wales, NSW 2052, Australia
Xueyuan Wu, Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia. E-mail: xueyuanw@unimelb.edu.au
David Pitt, Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, NSW 2109, Australia
Yan Wang, School of Mathematical and Geospatial Sciences, College of Science, Engineering and Health RMIT University, VIC 3000, Australia

References

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Bermúdez, L., Karlis, D. (2011). Bayesian multivariate Poisson models for insurance ratemaking. Insurance: Mathematics and Economics, 48, 226236.
Bermúdez, L. (2009). A priori ratemaking using bivariate Poisson regression models. Insurance: Mathematics and Economics, 44, 135141.
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), 821846.
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), 577599.
Culver, W.J. (1966). On the existence and uniqueness of the real logarithm of a matrix. Proceedings of the American Mathematical Society, 17, 11461151.
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.
Latouche, G., Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling. ASA SIAM, Philadelphia.
Neuts, M.F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Johns Hopkins University Press, Baltimore.
Panjer, H. (1981). Recursive evaluation of a family of compound distributions. ASTIN Bulletin, 12, 2226.
Pinquet, J., Guillén, M., Bolancé, C. (2001). Long-range contagion in automobile insurance data: estimation and implications for experience rating. ASTIN Bulletin, 31(2), 337348.
Schröter, K.J. (1990). On a family of counting distributions and recursions for the related compound distributions. Scandinavian Actuarial Journal, 161175.
Sundt, B. (2002). Recursive evaluation of aggregate claims distributions. Insurance: Mathematics and Economics, 30, 297332.
Sundt, B. (2003). Some recursions for moments of compound distributions. Insurance: Mathematics and Economics, 33, 486496.
Sundt, B., Jewell, W.S. (1981). Further results on recursive evaluation of compound distributions. ASTIN Bulletin, 12, 2739.
Sundt, B., Vernic, R. (2009). Recursions for convolutions and compound distributions with insurance applications. Springer-Verlag, Heidelberg.
Willmot, G. (1988). Sundt and Jewell's family of discrete distributions. ASTIN Bulletin, 18, 1729.
Wu, X., Li, S. (2010). Matrix-form recursions for a family of compound distributions. ASTIN Bulletin, 40, 351368.

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Matrix-form Recursive Evaluation of the Aggregate Claims Distribution Revisited

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

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