Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T19:40:40.337Z Has data issue: false hasContentIssue false
  • English
  • Français

Experience rating with Poisson mixtures

Published online by Cambridge University Press:  16 July 2015

Garfield O. Brown  and
Winston S. Buckley
Garfield O. Brown
Affiliation:
Statistical Laboratory, Centre for Mathematical Sciences, Cambridge, CB3 0WB, UK
Winston S. Buckley*
Affiliation:
Mathematical Sciences, Bentley University, Waltham, MA 02452, USA
*
*Correspondence to: Winston S. Buckley, Mathematical Sciences, Bentley University, Waltham, MA 02452, USA. Tel: 781-891-2000; Fax: 781-891-2457; E-mail: winb365@hotmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a Poisson mixture model for count data to determine the number of groups in a Group Life insurance portfolio consisting of claim numbers or deaths. We take a non-parametric Bayesian approach to modelling this mixture distribution using a Dirichlet process prior and use reversible jump Markov chain Monte Carlo to estimate the number of components in the mixture. Unlike Haastrup, we show that the assumption of identical heterogeneity for all groups may not hold as 88% of the posterior probability is assigned to models with two or three components, and 11% to models with four or five components, whereas models with one component are never visited. Our major contribution is showing how to account for both model uncertainty and parameter estimation within a single framework.

Keywords

Experience ratingPoisson mixturesReversible jumpMCMC
Type
Papers
Information
Annals of Actuarial Science , Volume 9 , Issue 2 , September 2015 , pp. 304 - 321
Copyright
© Institute and Faculty of Actuaries 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anastasiadis, S. & Chukova, S. (2012). Multivariate insurance models: an overview. Insurance: Mathematics and Economics, 51(1), 222–227.Google Scholar
Bermúdez, L. & Karlis, D. (2011). Bayesian multivariate Poisson models for insurance ratemaking. Insurance: Mathematics and Economics, 48(2), 226–236.Google Scholar
Bermúdez, L. & Karlis, D. (2012). A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking. Computational Statistics and Data Analysis, 56(12), 3988–3999.Google Scholar
Brooks, S.P. & Giudici, P. (1999). Diagnosing convergence of reversible jump MCMC algorithms. In J.M. Bernardo, J.O. Berger, A.P. Dawid & A.F.M. Smith, (Eds.), Bayesian Statistics 6 (pp. 733–742). Oxford University Press, Oxford.Google Scholar
Brooks, S.P., Giudici, P. & Philippe, A. (2003 a). Nonparametric convergence assessment for MCMC model selection. Journal of Computational and Graphical Statistics, 12, 1–22.Google Scholar
Brooks, S.P., Giudici, P. & Roberts, G.O. (2003 b). Efficient construction of reversible jump MCMC proposal distributions (with discussion). Journal of the Royal Statistical Society, Series B, 65(1), 3–55.Google Scholar
Carlin, B.P. & Chib, S. (1995). Bayesian model choice via Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, Series B, 57, 473–484.Google Scholar
Casella, G., Robert, C.P. & Wells, M.T. (2000). Mixture models, latent variables and partitioned importance sampling, technical report no. 2000–03, CREST, INSEE, Paris.Google Scholar
Dellaportas, P., Forster, J.J. & Ntzoufras, I. (2000). Bayesian variable selection using the Gibbs sampler. In D.K. Dey, S. Ghosh & B. Mallick (Eds.), Generalized Linear Models: A Bayesian Perspective (pp. 271--286). New York: Marcel Dekker.Google Scholar
Dellaportas, P., Forster, J.J. & Ntzoufras, I. (2003). On Bayesian model and variable selection using MCMC. Statistics and Computing, 12, 27–36.Google Scholar
Dellaportas, P. & Karlis, D. (2001). A simulation approach to nonparametric empirical Bayes analysis. International Statistical Review, 69(1), 63–79.Google Scholar
Dellaportas, P., Karlis, D. & Xekalaki, E. (2011). Bayesian analysis of finite Poisson mixtures. British Journal of Science, 1(1), 96–110.Google Scholar
Donnelly, C. & Wüthrich, M. (2012). Bayesian prediction of disability insurance frequencies using economic indicators. Annals of Actuarial Science, 6(2), 381–400.Google Scholar
England, P.D., Verrall, R.J., & Wüthrich, M.V. (2012). Bayesian Overdispersed Poisson Model and the Bornhuetter-Ferguson Claim Reserving Method. Annals of Actuarial Science, 6(2), 258–283.Google Scholar
Frühwirth-Schnatter, S. (2006). Finite Mixture and Markov Switching Models. Springer Series in Statistics. New York: Springer, ISBN-10: 0-387-32909-9.Google Scholar
Green, P.J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4), 711–732.Google Scholar
Green, P.J. & Richardson, S. (2001). Modelling heterogeneity with and without the Dirichlet process. Scandinavian Journal of Statistics, 28(2), 355–375.Google Scholar
Green, P.J. & Richardson, S. (2002). Hidden Markov models and disease mapping. Journal of the American Statistical Society, 97(460), 1055–1070.Google Scholar
Haastrup, S. (2000). Comparison of some Bayesian analyses of heterogeneity in group life insurance. Scandinavian Actuarial Journal, 2000, 2–16.CrossRefGoogle Scholar
Jewell, W.S. (1974). Credible means are exact Bayesian for exponential families. ASTIN Bulletin, 8(1), 77–90.Google Scholar
Katsis, A. & Ntzoufras, I. (2005). Bayesian hypothesis testing for the distribution of insurance claim counts using the Gibbs sampler. Journal of Computational Methods in Science and Engineering, 5, 201–214.Google Scholar
McLachlan, G. & Peel, D. (2000). Finite Mixture Models. Wiley-Interscience.Google Scholar
Norberg, R. (1989). Experience rating in group life insurance. Scandinavian Actuarial Journal, 1989, 194–224.Google Scholar
Ntzoufras, I. & Dellaportas, P. (2002). Bayesian modelling of outstanding liabilities incorporating claim count uncertainty. North American Actuarial Journal, 6(1), 113–128.Google Scholar
Ntzoufras, I., Katsis, A. & Karlis, A. (2005). Bayesian assessment of the distribution of insurance claim counts using the reversible jump algorithm. North American Actuarial Journal, 9, 90–105.Google Scholar
Phillips, D.B. & Smith, A.F.M. (1996). Bayesian model comparison via jump diffusions. In. W.R. Gilks, S. Richardson & D.J. Spiegelhalter (Eds.), Markov Chain Monte Carlo in Practice (pp. 215–239). Chapman and Hall.Google Scholar
Puustelli, A., Koskinen, L. & Luoma, A. (2008). Bayesian modelling of financial guarantee insurance. Insurance: Mathematics and Economics, 43(2), 245–254.Google Scholar
Robert, C.P. & Casella, G. (1999). Monte Carlo Statistical Methods. Springer.CrossRefGoogle Scholar
Stephens, M. (2000). Bayesian analysis of mixture models with an unknown number of components – an alternative to reversible jump methods. Annals of Statistics, 28(1), 40–74.Google Scholar
Streftaris, G. & Worton, B. (2008). Efficient and accurate approximate Bayesian inference with an application to insurance data. Computational Statistics and Data Analysis, 52(5), 2604–2622.Google Scholar
Titterington, D.M., Smith, A.F.M. & Makov, U.E. (1990). Statistical Analysis of Finite Mixture Distributions. Wiley, New York, NY.Google Scholar
Tremblay, L. (1992). Using the Poisson inverse Gaussian in bonus-malus systems. ASTIN Bulletin, 22(1), 97–106.CrossRefGoogle Scholar
Verrall, R.J., & Wüthrich, M.V. (2012). Reversible jump Markov chain Monte Carlo method for parameter reduction in claims reserving. North American Actuarial Journal, 16(2), 240–259.Google Scholar
Viallefont, V., Richardson, S. & Green, P.J. (2002). Bayesian analysis of Poisson mixtures. Journal of Nonparametric Statistics, 14(1–2), 181–202.Google Scholar
Walhin, J.F. & Paris, J. (1999). Using mixed Poisson processes in connection with bonus-malus systems. ASTIN Bulletin, 29(1), 81–99.Google Scholar
Walhin, J.F. & Paris, J. (2000). The true claim amount and frequency distributions within a bonus-malus system. ASTIN Bulletin, 30(2), 391–403.Google Scholar