Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-12T21:57:56.970Z Has data issue: false hasContentIssue false
  • English
  • Français

Model Uncertainty in Claims Reserving within Tweedie's Compound Poisson Models

Published online by Cambridge University Press:  09 August 2013

Gareth W. Peters ,
Pavel V. Shevchenko  and
Mario V. Wüthrich
Gareth W. Peters
Affiliation:
CSIRO Mathematical and Information Sciences, Sydney, Locked Bag 17, North Ryde, NSW, 1670, Australia UNSW Mathematics and Statistics Department, Sydney, 2052, Australia., E-Mail: peterga@maths.unsw.edu.au
Mario V. Wüthrich
Affiliation:
ETH Zurich, Department of Mathematics, CH-8092 Zurich, Switzerland., E-Mail: wueth@math.ethz.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we examine the claims reserving problem using Tweedie's compound Poisson model. We develop the maximum likelihood and Bayesian Markov chain Monte Carlo simulation approaches to fit the model and then compare the estimated models under different scenarios. The key point we demonstrate relates to the comparison of reserving quantities with and without model uncertainty incorporated into the prediction. We consider both the model selection problem and the model averaging solutions for the predicted reserves. As a part of this process we also consider the sub problem of variable selection to obtain a parsimonious representation of the model being fitted.

Keywords

Claims reservingmodel uncertaintyTweedie's compound Poisson modelBayesian analysismodel selectionmodel averagingMarkov chain Monte Carlo
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 39 , Issue 1 , May 2009 , pp. 1 - 33
Copyright
Copyright © International Actuarial Association 2009

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

Atchade, Y. and Rosenthal, J. (2005) On adaptive Markov chain Monte Carlo algorithms. Bernoulli 11(5), 815–828.CrossRefGoogle Scholar
Bedard, M. and Rosenthal, J.S. (2008) Optimal scaling of Metropolis algorithms: heading towards general target distributions. The Canadian Journal of Statistics 36(4), 483–503.CrossRefGoogle Scholar
Bernardo, J.M. and Smith, A.F.M. (1994) Bayesian Theory. John Wiley and Sons, NY.CrossRefGoogle Scholar
Cairns, A.J.G. (2000) A discussion of parameter and model uncertainty in insurance. Insurance: Mathematics and Economics 27, 313–330.Google Scholar
Carlin, B. and Chib, S. (1995) Bayesian model choice via Markov chain Monte Carlo methods. Journal of the Royal Statististical Society Series B 57, 473–484.Google Scholar
Casella, G. and George, E.I. (1992) Explaining the Gibbs Sampler. The American Statistician 46(3), 167–174.Google Scholar
Congdon, P. (2006) Bayesian model choice based on Monte Carlo estimates of posterior model probabilities. Computational Statistics and Data Analysis 50(2), 346–357.CrossRefGoogle Scholar
Dunn, P.K. and Smyth, G.K. (2005) Series evaluation of Tweedie exponential dispersion model densities. Statistics and Computing 15, 267–280.CrossRefGoogle Scholar
England, P.D. and Verrall, R.J. (2002) Stochastic claims reserving in general insurance. British Actuarial Journal 8(3), 443–510.CrossRefGoogle Scholar
Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. (1995) Bayesian Data Analysis. Chapman and Hall/CRC Texts in Statistical Science Series, 60.CrossRefGoogle Scholar
Gelman, A., Gilks, W.R. and Roberts, G.O. (1997) Weak convergence and optimal scaling of random walks metropolis algorithm. Annals of Applied Probability 7, 110–120.CrossRefGoogle Scholar
Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. (1996) Markov Chain Monte Carlo in Practice. Chapman and Hall, Florida.Google Scholar
Green, P. (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711–732.CrossRefGoogle 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
Robert, C.P. and Casella, G. (2004) Monte Carlo Statistical Methods, 2nd Edition Springer Texts in Statistics.CrossRefGoogle Scholar
Roberts, G.O. and Rosenthal, J.S. (2001) Optimal scaling for various Metropolis-Hastings algorithms. Statistical Science 16, 351–367.CrossRefGoogle Scholar
Rosenthal, J.S. (2007) AMCMC: An R interface for adaptive MCMC. Computational Statistics and Data Analysis 51(12), 5467–5470.CrossRefGoogle Scholar
Smith, A.F.M. and Roberts, G.O. (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. Journal of Royal Statistical Society Series B 55(1), 3–23.Google Scholar
Smyth, G.K. and Jørgensen, B. (2002) Fitting Tweedie's compound Poisson model to insurance claims data: dispersion modelling. Astin Bulletin 32, 143–157.CrossRefGoogle Scholar
Tweedie, M.C.K. (1984) An index which distinguishes between some important exponential families. In Statistics: Applications in new directions. Proceeding of the Indian Statistical Institute Golden Jubilee International Conference, Ghosh, J.K. and Roy, J. (eds.), 579–604, Indian Statistical Institute Canada.Google Scholar
Wright, E.M. (1935) On asymptotic expansions of generalized Bessel functions. Proceedings of London Mathematical Society 38, 257–270.CrossRefGoogle Scholar
Wüthrich, M.V. (2003) Claims reserving using Tweedie's compound Poisson model. Astin Bulletin 33, 331–346.CrossRefGoogle Scholar
Wüthrich, M.V. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance, Wiley Finance.Google Scholar