Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T15:18:58.615Z Has data issue: false hasContentIssue false
  • English
  • Français

Robust Bayesian Credibility Using Semiparametric Models

Published online by Cambridge University Press:  29 August 2014

Virginia R. Young
Virginia R. Young*
Affiliation:
University of Wisconsin– Madison
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In performing Bayesian analysis of insurance losses, one usually chooses a parametric conditional loss distribution for each risk and a parametric prior distribution to describe how the conditional distributions vary across the risks. Young (1997) applies techniques from nonparametric density estimation to estimate the prior and uses the estimated model to calculate the predictive mean of future claims given past claims. A shortcoming of this method is that, in estimating the prior, one assumes the average claim amount equals the conditional claim. In this paper, we consider a class of priors obtained by perturbing the one determined nonparametrically, as in Young (1997). We thereby reflect the uncertainty in the prior that arises from the randomness in the claim data. We, then, calculate intervals for the corresponding predictive means. We illustrate our method with data from Dannenburg et al. (1996) and compare the intervals of the predictive means with nonparametric confidence intervals.

Keywords

Kernel density estimationclaim estimationrobust Bayesian estimationDempster-Shafer belief functionsupper and lower expectations
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 28 , Issue 2 , November 1998 , pp. 187 - 203
Copyright
Copyright © International Actuarial Association 1998

References

Berger, J.O. (1994), An overview of robust Bayesian analysis, with discussion, Test, 3: 5–124.CrossRefGoogle Scholar
Bühlmann, H. (1967), Experience rating and credibility, ASTIN Bulletin, 4: 199–207.CrossRefGoogle Scholar
Bühlmann, H. (1970), Mathematical Models in Risk Theory, Springer-Verlag, New York.Google Scholar
Bühlmann, H. and Straub, E. (1970), Glaubwürdigkeit für Schadensätze, Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker, 70: 111–133.Google Scholar
Dannenburg, D.R., Kaas, R. and Goovaerts, M.J. (1996), Practical Actuarial Credibility Models, Institute of Actuarial Science and Econometrics, University of Amsterdam, Amsterdam, The Netherlands.Google Scholar
Dempster, A.P. (1967), Upper and lower probabilities induced by a multivalued mapping, Annuals of Mathematical Statistics, 38: 325–339.CrossRefGoogle Scholar
Dempster, A.P. (1968), A generalization of Bayesian inference, with discussion, Journal of the Royal Statistical Society, Series B, 30: 205–247.Google Scholar
Denneberg, D. (1994), Non-Additive Measure and Integral, Kluwer, Dordrecht.CrossRefGoogle Scholar
Frees, E.W., Young, V.R. and Luo, Y. (1998), A longitudinal data analysis interpretation of credibility models, working paper.Google Scholar
Sereling, R.J. (1980), Approximation Theorems of Mathematical Statistics, Wiley, New York.CrossRefGoogle Scholar
Shafer, G. (1979), Allocations of probability, Annals of Probability, 7: 827–839.CrossRefGoogle Scholar
Silverman, B.W. (1986), Density Estimation for Statistics and Data Analysis, Chapman & Hall, London.Google Scholar
Walley, P. (1991), Statistical Reasoning with Imprecise Probabilities, Chapman & Hall, London.CrossRefGoogle Scholar
Wasserman, L.A. (1990a), Belief functions and statistical inference, Canadian Journal of Statistics, 18: 183–196.CrossRefGoogle Scholar
Wasserman, L.A. (1990b), Prior envelopes based on belief functions, Annals of Statistics, 18: 454–464.CrossRefGoogle Scholar
Young, V.R. (1997), Credibility using semiparametric models, ASTIN Bulletin, 27: 273–285.CrossRefGoogle Scholar