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Bayesian Nonparametric Approach to Credibility Modelling

Published online by Cambridge University Press:  10 May 2011

A. Gangopadhyay  and
W.-C. Gau
A. Gangopadhyay
Affiliation:
Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, U.S.A., Email: [email protected]
W.-C. Gau
Affiliation:
Department of Statistics and Actuarial Science, University of Central Florida, Orlando, FL 32816, U.S.A., Email: [email protected]
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Abstract

Current methods in credibility theory often rely on parametric models, e.g., a linear function of past experience. During the last decade, the existence of high speed computers and statistical software packages allowed the introduction of more sophisticated and flexible modelling strategies. In recent years, some of these techniques, which made use of the Markov Chain Monte Carlo (MCMC) approach to modelling, have been incorporated in credibility theory. However, very few of these methods made use of additional covariate information related to risk, or collection of risks; and at the same time account for the correlated structure in the data. In this paper, we consider a Bayesian nonparametric approach to the problem of risk modelling. The model incorporates past and present observations related to risk, as well as relevant covariate information. This Bayesian modelling is carried out by sampling from a multivariate Gaussian prior, where the covariance structure is based on a thin-plate spline. The model uses the MCMC technique to compute the predictive distribution of future claims based on available data.

Keywords

Thin-Plate SplineMarkov Chain Monte Carlo (MCMC)Loss ModelsCredibility
Type
Papers
Information
Annals of Actuarial Science , Volume 2 , Issue 1 , March 2007 , pp. 91 - 114
Copyright
Copyright © Institute and Faculty of Actuaries 2007

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References

Bühlmann, H. (1967). Experience rating and credibility. ASTIN Bulletin, 4, 199–207.Google Scholar
Bühlmann, H. & Straub, E. (1970). Glaubwurdigkeit fur Schadensatze. Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker, 70, 111–133.Google Scholar
Duchon, J. (1975). Functions splines et vecteurs aleatoires. Technical Report 213, Seminaire d'Analyse Numeriaue.Google Scholar
Duchon, J. (1976). Functions-spline et esperances conditionnelles de champs Gaussians Annal Science University Clermont Ferrand II Mathematics, 14, 19–27.Google Scholar
Duchon, J. (1977). Spline minimising rotation-invariant semi-norms in Sobolev spaces in constructive theory of functions of several variables. Springer-Verlag, Berlin.Google Scholar
Gangopadhyay, A. & Gau, W. (2003). Credibility modeling via spline nonparametric regression. CAS Forum, Winter, 215–252.Google Scholar
Gelfand, A.E. & Smith, A.F.M. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398–409.CrossRefGoogle Scholar
Hastie, T.J. & Tibshirani, R.J. (1990). Generalized additive models. Chapman & Hall.Google Scholar
Jewell, W.S. (1974). Credible means are exact Bayesian for exponential families. ASTIN Bulletin, 8, 336–341.Google Scholar
Klugman, S.A. (1992). Bayesian statistics in actuarial science with emphasis on credibility. Kluwer, Boston.CrossRefGoogle Scholar
Mayerson, A.L. (1964). A Bayesian view of credibility. Proceedings of the Casualty Actuarial Society, 51, 85–104.Google Scholar
Meinguet, J. (1979). Multivariate interpolation at arbitrary points made simple. Journal of Applied Mathematical Physics (ZAMP), 30, 292–304.Google Scholar
Scollnik, D.P.M. (2000). Actuarial modeling with MCMC and BUGS. Actuarial Education and Research.Google Scholar
Spiegelhalter, D.J., Thomas, A. & Best, N.G. (1996). BUGS 0.5: Bayesian inference using Gibbs sampling manual (version II). MRC Biostatistics Unit, Cambridge.Google Scholar
Wahba, G. & Wendelberger, J. (1980). Some new mathematical methods for variational objective analysis using splines and cross-validation. Monthly Weather Review, 108, 1122–1145.Google Scholar
Wahba, G. (1990). Spline models for observational data. SIAM, Philadelphia, 69.CrossRefGoogle Scholar
Wahba, G. (2000). An introduction to model building with reproducing kernel Hilbert space. Technical report No. 1020, Department of Statistics, University of Wisconsin, Madison, WI.Google Scholar
Wong, C. & Kohn, R. (1996). A Bayesian approach to additive semiparametric regression. Journal of Econometrics, 7, 209–223.CrossRefGoogle Scholar
Wood, S., Kohn, R., Shively, T. & Jiang, W. (2002). Model selection in spline nonparametric regression. Journal of the Royal Statistical Society B, 64, 119–139.CrossRefGoogle Scholar
Wood, S. N. (2003). Thin plate regression splines. Journal of the Royal Statistical Society B, 65, Part 1, 95–114.CrossRefGoogle Scholar
Young, V.R. (1997). Credibility using a loss function from spline theory. Scandinavian Actuarial Journal, 2, 160–185.Google Scholar
Young, V.R. (2000). Robust Bayesian credibility using semiparametric models. Preprint.Google Scholar