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Robust Bayesian Analysis of Loss Reserves Data Using the Generalized-t Distribution

Published online by Cambridge University Press:  17 April 2015

Jennifer S.K. Chan ,
S.T. Boris Choy  and
Udi E. Makov
Jennifer S.K. Chan
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia, E-Mail: [email protected]
S.T. Boris Choy
Affiliation:
Department of Mathematical Sciences, University of Technology Sydney, P.O. Box 123, Broadway, NSW 2007, Australia, E-Mail: [email protected]
Udi E. Makov
Affiliation:
Department of Statistics, University of Haifa, Haifa, 31905 Israel, E-Mail: [email protected]
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Abstract

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This paper presents a Bayesian approach using Markov chain Monte Carlo methods and the generalized-t (GT) distribution to predict loss reserves for the insurance companies. Existing models and methods cannot cope with irregular and extreme claims and hence do not offer an accurate prediction of loss reserves. To develop a more robust model for irregular claims, this paper extends the conventional normal error distribution to the GT distribution which nests several heavy-tailed distributions including the Student-t and exponential power distributions. It is shown that the GT distribution can be expressed as a scale mixture of uniforms (SMU) distribution which facilitates model implementation and detection of outliers by using mixing parameters. Different models for the mean function, including the log-ANOVA, log-ANCOVA, state space and threshold models, are adopted to analyze real loss reserves data. Finally, the best model is selected according to the deviance information criterion (DIC).

Keywords

Bayesian approachstate space modelthreshold modelscale mixtures of uniform distributiondeviance information criterion
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 38 , Issue 1 , May 2008 , pp. 207 - 230
Copyright
Copyright © ASTIN Bulletin 2008

References

Akaike, H. (1974) A new look at the statistical model identification, IEEE Transactions on Automatic Control, 19, 716–723.CrossRefGoogle Scholar
Arslan, O. and Genc, A.I. (2003) Robust location and scale estimation based on the univariate generalized t (GT) distribution, Communications in Statistics: Theory and Methods, 32, 1505–1525.CrossRefGoogle Scholar
Box, G.E.P. and Tiao, G.C. (1973) Bayesian Inference in Statistical Analysis. Mass.: Addison-Wesley.Google Scholar
Butler, R.J., Mcdonald, B.J., Nelson, R.D. and White, S.B. (1990) Robust and Partially Adaptive Estimation of Regression Models, MIT Press, 72, 321–327.Google Scholar
Choy, S.T.B. and Chan, C.M. (2003) Scale mixtures distributions in insurance applications, ASTIN Bulletin, 33, 93–104.CrossRefGoogle Scholar
Choy, S.T.B. and Smith, A.F.M. (1997) Hierarchical models with scale mixtures of normal distribution, TEST, 6, 205–221.CrossRefGoogle Scholar
Dawid, A.P. (1973) Posterior expectations for large observations, Biometrika, 60, 664–666.CrossRefGoogle Scholar
De Alba, E. (2002) Bayesian estimation of outstanding claim reserves, North American Actuarial Journal, October.CrossRefGoogle Scholar
De Jong, P. and Zehnwirth, B. (1983) Claims reserving state space models and Kalman filter, The Journal of the Institute of Actuaries, 110, 157–181.CrossRefGoogle Scholar
Damien, P., Wakefield, J. and Walker, S. (1999) Gibbs sampling for Bayesian non-Conjugate and hierarchical models by using auxiliary variables, Journal of the Royal Statistical Society, Series B 61, 331–344.Google Scholar
Gelfand, A.E. and Smith, A.F.M. (1990) Sampling-based approaches to calculating marginal densities, Journal of American Statistical Association, 85, 398–409.CrossRefGoogle Scholar
Geman, S. and Geman, D. (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.CrossRefGoogle ScholarPubMed
Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. (2004) Bayesian Data Analysis, 2nd edition, Chapman & Hall/CRC, 182–184.Google Scholar
Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. (1998) Markov chain Monte Carlo in practice, Boca Raton, Fla.: Chapman and Hall.Google Scholar
Green, P.J. (1995) Reversible jump Markoc chain Monte Carlo computation and Bayesian model determination, Biometrika, 82, 711–732.CrossRefGoogle Scholar
Hager, H.W. and Bain, L.J. (1970) Inferential procedures for the generalized gamma distribution, Journal of the American Statistical Association, 65, 1601–1609.CrossRefGoogle Scholar
Hastings, W.K. (1970) Monte Carlo sampling methods using Markov Chains and their applications, Biometrika, 57, 97–10.CrossRefGoogle Scholar
Hazan, A. and Makov, U.E. (2001) A switching regression model for loss reserves, Technical report TR24.Google Scholar
Makov, U.E. (2001) Perspective applications of Bayesian methods in actuarial science: a perspective, North American Actuarial Journal, 5.CrossRefGoogle Scholar
Meng, X.L. (1994) Posterior predictive p-values, The Annual of Statistics, 22.CrossRefGoogle Scholar
McDonald, J.B. and Butler, R.J. (1987) Some generalized mixture distributions with an application to unemployment duration, Review of Economics and Statistics, 69, 232–240.CrossRefGoogle Scholar
McDonald, J.B. and Newey, W.K. (1988) Partially adaptive estimation of regression models via the generalized t distribution, Economic Theory, 4, 428–457.CrossRefGoogle Scholar
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N. and Teller, A.H. (1953) Equations of State calculations by fast computing machines, Journal of Chemical Physics, 21, 1087–1091.CrossRefGoogle Scholar
Ntzoufras, I. and Dellaportas, P. (2002) Bayesian modeling of outstanding liabilities incorporating claim count uncertainty, North American Actuarial Journal, 6, 113–128.CrossRefGoogle Scholar
Ntzoufras, I., Katsis, A. and Karlis, D. (2005) Bayesian assessment of the distribution of insurance claim counts using reversible jump MCMC, North American Actuarial Journal, 9, 90–108.CrossRefGoogle Scholar
Renshaw, A.E. (1989) Chain ladder and interactive modeling, The Journal of the Institute of Actuaries, 116, 559–587.CrossRefGoogle Scholar
Renshaw, A.E. and Verrall, R.J. (1998) A stochastic model underlying the chain-ladder technique, British Actuarial Journal, 4, 903–926.CrossRefGoogle Scholar
Schwarz, G. (1978) Estimating the dimension of a model, Annals of Statistics 6, 461–464.CrossRefGoogle Scholar
Scollnik, D.P.M. (1998) On the analysis of truncated generalized Poisson distribution using a Bayesian method, ASTIN Bulletin, 28, 135–152.CrossRefGoogle Scholar
Scollnik, D.P.M. (2001) Actuarial modeling with MCMC and BUGS, North American Actuarial Journal, 5, 96–124.CrossRefGoogle Scholar
Scollnik, D.P.M. (2002) Implementation of four models for outstanding liabilities in WinBUGS: A discussion of a paper by Ntzoufras and Dellaportas, North American Actuarial Journal, 6, 128–136.CrossRefGoogle Scholar
Spiegelhalter, D., Best, N.G., Carlin, B.P. and van der Linde, A. (2002) Bayesian measures of model complexity and fit, Journal of the Royal Statistical Society, Series B, 64, 583–639.CrossRefGoogle Scholar
Spiegelhalter, D., Thomas, A., Best, N.G. and Lunn, D. (2004) Bayesian inference using Gibbs sampling for Window version (WinBUGS), Version 1.4.1, MRC Biostatistics Unit, Institute of Public health, Cambridge, UK. (www.mrc-bsu.cam.ac.uk/bugs/welcome.shtml).Google Scholar
Verrall, R.J. (1989) State space representation of the chain ladder linear model, The Journal of Institute of Actuaries, 116, 589–610.CrossRefGoogle Scholar
Verrall, R.J. (1991) Chain ladder and maximum likelihood, The Journal of Institute of Actuaries, 118, 489–499.CrossRefGoogle Scholar
Verrall, R.J. (1994) A method of modeling varying-off evolutions in claims reserving, ASTIN Bulletin, 24, 325–332.CrossRefGoogle Scholar
Verrall, R.J. (1996) Claims reserving and generalized additive models, The Journal of Institute of Actuaries, 19, 31–43.Google Scholar
Verrall, R.J. (2004) A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving, North American Actuarial Journal, 8, 67–89.CrossRefGoogle Scholar
Walker, S.G. and Gutiérrez-Peña, E. (1999) Robustifying Bayesian procedures, In: Bayesian Statistics 6. Oxford, New York, 685–710.Google Scholar