Abstract
In this paper we consider statistical problems arising from applications concerning insurance-premium calculation. We describe an integrated set of Bayesian tools for modelling bonus-malus systems (BMS) for insurance premiums. This paper describes a bonus-malus system (BMS) applicable to insurance claims procedures, constructed using a hierarchical Bayesian model. We then address notions and techniques of robust Bayesian analysis in the context of problems arising in BMS.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berger, J. O. (1985),Statistical Decision Theory and Bayesian Analysis, Springer-Verlag, New York.
Berger, J. O. (1994). An overview of robust Bayesian analysis (with discussion).Test, 3(1):5–124.
Berger, J. O., Ríos, D., andRucgeri, F. (2000). Bayesian robustness. In D. Ríos and F. Ruggeri, eds.,Robust Bayesian Analysis, pp. 1–32. Springer-Verlag, New York.
Betrò, B., Ruggeri, F., andMęczarski, M. (1994). Robust Bayesian analysis under generalized moments conditions.Journal of Statistical Planning and Inference, 41:257–266.
Bühlmann, H. (1970).Mathematical Methods in Risk Theory, Springer-Verlag, New York.
Eichenauer, J., Lehn, J., andRettig, S. (1988). A gamma-minimax result in credibility theory.Insurance: Mathematics & Economics, 7:49–57.
Frangos, N. andVrontos, S. (2001). Design of optimal bonus-malus systems with a fequency and severity component on an individual basis in automobile insurance.ASTIN Bulletin, 31:1–22.
Gómez, E., Hernández, A., andVázquez-Polo, F. J. (2000). Robust Bayesian premium principles in actuarial science.Journal of the Royal Statistical Society. Series D (The Statistician), 49:241–252.
Gómez, E., Pérez, J. M., Hernández, A, andVázquez-Polo, F. J. (2002). Measuring sensitivity in a bonus-malus system.Insurance: Mathematics & Economics, 31:105–113.
Goovaerts, M. andHoogstad, W. (1987). Credibility theory.Surveys of Actuarial Studies, 4:19–52.
Heilmann, W. (1989). Decision theoretic foundations of credibility theory.Insurance: Mathematics & Economics, 8:77–95.
Klugman, S. (1992).Bayesian Statistics in Acturial Science. Kluwer Academic Publisher, Boston.
Lemaire, J. (1995).Bonus-Malus Systems in Automobile Insurance. Kluwer Academic Publisher, London.
Magalhães, F. andDunsmore, I. R (2003). Predicting the number of accidents at a road junction.Test, 12(1):153–172.
Makov, U. (1995). Loss robustness via Fisher-weighted squared error loss function.Insurance: Mathematics & Economics, 16:1–6.
Martín, J., Ríos, D., andRuggeri, F. (2003). Joint sensitivity in Bayesian decision theory.Test, 12(1):173–194.
Miller, R. B. (1980). Actuarial applications of Bayesian statistics. In A. Zellner, ed.,Bayesian Analysis in Econometrics and Statistics, pp. 197–212. North-Holland, New York.
Ntzoufras, I. andDellaportas, P. (2002). Bayesian modelling of out-standing liabilities incorporating claim count uncertainty.North American Actuarial Journal, 6:113–128.
Ríos, S., Martín, J., Ríos, D. andRuggeri, F. (1999). Bayesian forecasting for accident proneness evaluation.Scandinavian Actuarial Journal, 2:134–156.
Tremblay, L. (1992). Using the Poisson inverse Gaussian in bonus-malus systems.ASTIN Bulletin, 22:97–106.
Winkler, G. (1988). Extreme points of moment sets.Mathematical Operations Research, 13:581–587.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by grants from MCyT (Ministerio de Ciencia y Tecnología, Spain, project BEC2001-3774) and DGUI (Dirección General de Universidades e Investigación del Gobierno Autónomo de Canarias, Spain, project PI2003-033).
Rights and permissions
About this article
Cite this article
Gómez-Déniz, E., Vázquez-Polo, F.J. & Pérez, J.M. A note on computing bonus-malus insurance premiums using a hierarchical bayesian framework. Test 15, 345–359 (2006). https://doi.org/10.1007/BF02607056
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02607056