Experience rating with Poisson mixtures

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.

[1]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[2]  S. Chukova,et al.  Multivariate insurance models: An overview , 2012 .

[3]  Dimitris Karlis,et al.  Bayesian Assessment of the Distribution of Insurance Claim Counts Using Reversible Jump MCMC , 2005 .

[4]  B. Carlin,et al.  Bayesian Model Choice Via Markov Chain Monte Carlo Methods , 1995 .

[5]  Petros Dellaportas,et al.  On Bayesian model and variable selection using MCMC , 2002, Stat. Comput..

[6]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[7]  P. Dellaportas,et al.  Bayesian variable selection using the Gibbs sampler , 2000 .

[8]  M. Stephens.  Bayesian analysis of mixture models with an unknown number of components- an alternative to reversible jump methods , 2000 .

[9]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[10]  G. Casella,et al.  Mixture models, latent variables and partitioned importance sampling , 2004 .

[11]  P. Green,et al.  Modelling Heterogeneity With and Without the Dirichlet Process , 2001 .

[12]  P. Green,et al.  Bayesian Analysis of Poisson Mixtures , 2002 .

[13]  Reversible Jump Markov Chain Monte Carlo Method for Parameter Reduction in Claims Reserving , 2012 .

[14]  P. Dellaportas,et al.  A Simulation Approach to Nonparametric Empirical Bayes Analysis , 2001 .

[15]  Ioannis Ntzoufras,et al.  Bayesian hypothesis testing for the distribution of insurance claim counts using the Gibbs sampler , 2005, J. Comput. Methods Sci. Eng..

[16]  Ragnar Norberg,et al.  Experience Rating in Group Life Insurance , 1989 .

[17]  Dimitris Karlis,et al.  Bayesian multivariate Poisson models for insurance ratemaking , 2011 .

[18]  Mario V. Wüthrich,et al.  Bayesian prediction of disability insurance frequencies using economic indicators , 2012, Annals of Actuarial Science.

[19]  Petros Dellaportas,et al.  Bayesian Modelling of Outstanding Liabilities Incorporating Claim Count Uncertainty , 2002 .

[20]  George Streftaris,et al.  Efficient and accurate approximate Bayesian inference with an application to insurance data , 2008, Comput. Stat. Data Anal..

[21]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[22]  Using Mixed Poisson Processes in Connection with Bonus-Malus Systems , 1999, ASTIN Bulletin.

[23]  P. Green.  Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[24]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[25]  O. Cappé,et al.  Reversible jump, birth‐and‐death and more general continuous time Markov chain Monte Carlo samplers , 2003 .

[26]  Arto Luoma,et al.  Bayesian modelling of financial guarantee insurance , 2008 .

[27]  P. Dellaportas,et al.  Bayesian Analysis of Finite Poisson Mixtures , 2011 .

[28]  Chirag Parikh,et al.  IDEA cryptographic processor in FPGA , 2005, J. Comput. Methods Sci. Eng..

[29]  P. Green,et al.  Hidden Markov Models and Disease Mapping , 2002 .

[30]  Bayesian over-dispersed Poisson model and the Bornhuetter & Ferguson claims reserving method , 2012, Annals of Actuarial Science.

[31]  G. Roberts,et al.  Efficient construction of reversible jump Markov chain Monte Carlo proposal distributions , 2003 .

[32]  Sylvia Frühwirth-Schnatter,et al.  Finite Mixture and Markov Switching Models , 2006 .

[33]  Luc Tremblay.  Using the Poisson Inverse Gaussian in Bonus-Malus Systems , 1992, ASTIN Bulletin.

[34]  Dimitris Karlis,et al.  A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking , 2012, Comput. Stat. Data Anal..

[35]  Walter R. Gilks,et al.  Bayesian model comparison via jump diffusions , 1995 .

[36]  Comparison of Some Bayesian Analyses of Heterogeneity in Group Life Insurance , 2000 .

[37]  William S. Jewell,et al.  Credible Means are exact Bayesian for Exponential Families , 1974, ASTIN Bulletin.