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Fitting Tweedie's Compound Poisson Model to Insurance Claims Data: Dispersion Modelling

Published online by Cambridge University Press:  29 August 2014

Gordon K. Smyth  and
Bent Jørgensen
Gordon K. Smyth*
Affiliation:
Walter and Eliza Hall Institute of Medical Research, Melbourne, Australia
Bent Jørgensen
Affiliation:
Department of Statistics and Demograph, University of Southern Denmark
*
1Dr G.K. Smyth, Bioinformatics, Walter and Eliza Hall Institute of Medical Research, Post Office, Royal Melbourne Hospital, Parkville, VIC 3050, Australia
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Abstract

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We reconsider the problem of producing fair and accurate tariffs based on aggregated insurance data giving numbers of claims and total costs for the claims. Jørgensen and de Souza (Scand Actuarial J., 1994) assumed Poisson arrival of claims and gamma distributed costs for individual claims. Jørgensen and de Souza (1994) directly modelled the risk or expected cost of claims per insured unit, μ say. They observed that the dependence of the likelihood function on μ is as for a linear exponential family, so that modelling similar to that of generalized linear models is possible. In this paper we observe that, when modelling the cost of insurance claims, it is generally necessary to model the dispersion of the costs as well as their mean. In order to model the dispersion we use the framework of double generalized linear models. Modelling the dispersion increases the precision of the estimated tariffs. The use of double generalized linear models also allows us to handle the case where only the total cost of claims and not the number of claims has been recorded.

Keywords

Car insuranceClaims dataCompound Poisson modelExposureGeneralized linear modelsDispersion modellingDouble generalized linear modelsPower variance functionREMLRisk theoryTarification

Information

Type
Workshop
Information
ASTIN Bulletin: The Journal of the IAA , Volume 32 , Issue 1 , May 2002 , pp. 143 - 157
Copyright
Copyright © International Actuarial Association 2002

References

Andrews, D.F., and Herzberg, A.M. (1985) Data. A collection of problems from many fields for the student and research worker. Springer-Verlag, New York, pp. 413–421.Google Scholar
Cox, D.R. and Reid, N. (1987) Parameter orthogonality and approximate conditional inference. J.R. Statist. Soc B 49, 1–39.Google Scholar
Dobson, A.J. (2001) An Introduction to Generalized Linear Models, Second Edition. Chapman and Hall/CRC, London.Google Scholar
Dunn, P.K. (2001) Likelihood-Based Inference for Tweedie Generalized Linear Models PhD Thesis, Department of Mathematics, University of Queensland, Brisbane, Australia.Google Scholar
Haberman, S., and Renshaw, A.E. (1998) Actuarial applications of generalized linear models. In Statistics in Finance, Hand, D.J. and Jacka, S.D. (eds), Arnold, London.Google Scholar
Hallin, M., and Ingenbleek, J.-F. (1983) The Swedish automobile portfolio in 1977. A statistical study. Scandinavian Actuarial Journal, 49–64.CrossRefGoogle Scholar
Jørgensen, B. (1987) Exponential dispersion models. J. R. Statist. Soc. B 49, 127–162.Google Scholar
Jørgensen, B. (1997) Theory of Dispersion Models. Chapman & Hall, London.Google Scholar
Jørgensen, B. and De Souza, M.C.P. (1994) Fitting Tweedie's compound Poisson model to insurance claims data. Scandinavian Actuarial Journal, 69–93.CrossRefGoogle Scholar
Lee, Y. and Nelder, J.A. (1998) Generalized linear models for the analysis of quality-improvement experiments. Canadian Journal of Statistics 26, 95–105.CrossRefGoogle Scholar
McCullagh, P. and Nelder, J.A. (1989) Generalized Linear Models, 2nd Edition. Chapman & Hall, London.CrossRefGoogle Scholar
Millenhall, S.J. (1999) A systematic relationship between minimum bias and generalized linear models. 1999 Proceedings of the Casualty Actuarial Society 86, 393–487.Google Scholar
Murphy, K.P., Brockman, M.J. and Lee, P.K.W. (2000) Using generalized linear models to build dynamic pricing systems. Casualty Actuarial Forum, Winter 2000.Google Scholar
Nelder, J.A., and Lee, Y. (1992) Likelihood, quasi-likelihood and pseudo-likelihood: some comparisons. J. R. Statist. Soc. B 54, 273–284.Google Scholar
Nelder, J.A. and Pregibon, D. (1987) An extended quasi-likelihood function. Biometrik 74, 221–231.CrossRefGoogle Scholar
Renshaw, A.E. (1994) Modelling the claims process in the presence of covariates. ASTIN Bulletin 24, 265–286.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V., and Teugels, J. (1999) Stochastic Processes for Insurance and Finance. John Wiley & Sons, Chichester.CrossRefGoogle Scholar
Smyth, G.K. (1989) Generalized linear models with varying dispersion. J. Roy. Statist. Soc. B 51, 47–60.Google Scholar
Smyth, G.K. (1996) Partitioned algorithms for maximum likelihood and other nonlinear estimation. Statistics and Computing 6, 201–216.CrossRefGoogle Scholar
Smyth, G.K., and Verbyla, A.P. (1999) Adjusted likelihood methods for modelling dispersion in generalized linear models. Environments 10, 696–709.Google Scholar
Smyth, G.K., Huele, F., and Verbyla, A.P. (2001) Exact and approximate REML for heteroscedastic regression. Statistical Modelling 1, 161–175.CrossRefGoogle Scholar