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Chain-Ladder as Maximum Likelihood Revisited

Published online by Cambridge University Press:  10 May 2011

D. Kuang ,
B. Nielsen  and
J. P. Nielsen
B. Nielsen
Affiliation:
Nuffield College, Oxford OX1 5AD, UK. :, Email: [email protected]
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Abstract

It has long been known that maximum likelihood estimation in a Poisson model reproduces the chain-ladder technique. We revisit this model. A new canonical parametrisation is proposed to circumvent the inherent identification problem in the parametrisation. The maximum likelihood estimators for the canonical parameter are simple, interpretable and easy to derive. The boundary problem where all observations in one particular development year or on particular underwriting year is zero is also analysed.

Keywords

Boundary ProblemCanonical ParameterChain-LadderIdentification ProblemMaximum LikelihoodPoisson Model
Type
Papers
Information
Annals of Actuarial Science , Volume 4 , Issue 1 , March 2009 , pp. 105 - 121
Copyright
Copyright © Institute and Faculty of Actuaries 2009

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References

Barndorff-Nielsen, O.E. (1978). Information and exponential families. Wiley, New York.Google Scholar
Cox, D.R. & Hinkley, D.V. (1974). Theoretical statistics. Chapman and Hall, London.Google Scholar
Eaton, M.L. (2007). Multivariate statistics: a vector space approach. Lecture Notes–Monograph Series 53. Beachwood, Institute of Mathematical Statistics, OH.CrossRefGoogle Scholar
England, P.D. & Verrall, R.J. (1999). Analytic and bootstrap estimates of prediction errors in claims reserving. Insurance: Mathematics and Economics, 25, 281–293.Google Scholar
England, P.D. & Verrall, R.J. (2002). Stochastic claims reserving in general insurance. British Actuarial Journal, 8, 519–544.Google Scholar
Hachemeister, C.A. & Stanard, J.N. (1975). IBNR claims count estimation with static lag functions. Unpublished manuscript.Google Scholar
Kremer, E. (1982). IBNR-claims and the two-way model of ANOVA. Scandinavian Actuarial Journal, 47–55.Google Scholar
Kremer, E. (1985). Einführung in die Versicherungsmathematik. Vandenhoek & Ruprecht, Gottingen.Google Scholar
Kuang, D., Nielsen, B. & Nielsen, J.P. (2008a). Identification of the age-period-cohort model and the extended chain-ladder model. Biometrika, 95, 979–986.Google Scholar
Kuang, D., Nielsen, B. & Nielsen, J.P. (2008b). Forecasting with the age-period-cohort model and the extended chain-ladder model. Biometrika, 95, 987–991.Google Scholar
Mack, T. (1991). A simple parametric model for rating automobile insurance or estimating IBNR claims reserves. ASTIN Bulletin, 21, 93–109.Google Scholar
Mack, T. & Venter, G. (2000). A comparison of stochastic models that reproduce chain-ladder reserve estimates. Insurance: Mathematics and Economics, 26, 101–107.Google Scholar
Rockafellar, R.T. (1970). Convex analysis. Princeton University Press, Princeton, NJ.Google Scholar
Verrall, R.J. (1991). Chain-ladder and maximum likelihood. Journal of the Institute of Actuaries, 118, 489–499.Google Scholar
Verrall, R.J. (1994). Statistical methods for the chain-ladder technique. Casualty Actuarial Society Forum, Spring 1994, 393–446.Google Scholar
Verrall, R.J. & England, P.D. (2000). Comments on “A comparison of stochastic models that reproduce chain-ladder reserve estimates”, by Mack and Venter. Insurance: Mathematics and Economics, 26, 109–111.Google Scholar