Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T07:15:40.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Posterior Regret Γ-Minimax Estimation of Insurance Premium in Collective Risk Model

Published online by Cambridge University Press:  17 April 2015

Agata Boratyńska
Agata Boratyńska*
Affiliation:
Institute of Econometrics, Warsaw School of Economics, Al. Niepodległošci 162, 02-554 Warszawa, Poland, E-mail: aborata@sgh.waw.pl
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.

The collective risk model for the insurance claims is considered. The objective is to estimate a premium which is defined as a functional H specified up to an unknown parameter θ (the expected number of claims). Four principles of calculating a premium are applied. The Bayesian methodology, which combines the prior knowledge about a parameter θ with the knowledge in the form of a random sample is adopted. Two loss functions (the square-error loss function and the asymmetric loss function LINEX) are considered. Some uncertainty about a prior is assumed by introducing classes of priors. Considering one of the concepts of robust procedures the posterior regret Γ-minimax premiums are calculated, as an optimal robust premiums. A numerical example is presented.

Keywords

Bayesian modelclasses of priorsposterior regretsquare error lossLINEXinsurance premium
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 38 , Issue 1 , May 2008 , pp. 277 - 291
Copyright
Copyright © ASTIN Bulletin 2008

References

Berger, J.O. (1994) An overview of robust Bayesian analysis. Test 3, 5–124 (with discussion).CrossRefGoogle Scholar
Boratyńska, A. (2002a) Posterior regret г-minimax estimation in a normal model with asymmetric loss function. Applicationes Mathematicae 29, 7–13.CrossRefGoogle Scholar
Boratynska, A. (2002b) Robust Bayesian estimation with asymmetric loss function. Applicationes Mathematicae 29, 297–306.CrossRefGoogle Scholar
Gómez-Déniz, E., Hernandez-Bastida, A. and Vazquez-Polo, F.J. (1999) The Esscher premium principle in risk theory: a Bayesian sensitivity study. Insurance: Mathematics and Economics 25, 387–395.Google Scholar
Gómez-Déniz, E., Hernandez, A., Pérez, J.M. and Vazquez-Polo, F.J. (2002a) Measuring sensitivity in a bonus-malus system. Insurance: Mathematics and Economics 31, 105–113.Google Scholar
Gómez-Déniz, E., Hernandez-Bastida, A. and Vázquez-Polo, F.J. (2002b) Bounds for ratios of posterior expectations: applications in the collective risk model. Scand. Actuarial J., 37–44.CrossRefGoogle Scholar
Gómez-Déniz, E., Pérez, J.M. and Vázquez-Polo, F.J. (2006) On the use of posterior regret Gamma-minimax actions to obtain credibility premiums. Insurance: Mathematics and Economics 39, 115–121.Google Scholar
Heilmann, W.R. (1989) Decision theoretic foundations of credibility theory. Insurance: Mathematics and Economics 8, 77–95.Google Scholar
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2001) Modern Actuarial Risk Theory. Kluwer Academic Publishers, Boston.Google Scholar
Klugman, S. (1992) Bayesian Statistics in Actuarial Science. Kluwer Academic Publisher, Boston.CrossRefGoogle Scholar
Klugman, S., Panjer, H. and Willmot, G. (1998) Loss Model from Data to Decisions. Wiley, New York.Google Scholar
Lavine, M., Wasserman, L. and Wolpert, R.L. (1991) Bayesian inference with specified priors marginals. JASA 86, 964–971.CrossRefGoogle Scholar
Lemaire, J. (1979) How to define a bonus-malus system with an exponential utility function. Astin Bulletin 10, 274–282.CrossRefGoogle Scholar
Makov, U., Smith, A., Liu, Y.-H., et al. (1996) Bayesian methods in actuarial science. The Statistician 45, 503–515.CrossRefGoogle Scholar
Ríos Insua, S., Martin, J., Ríos Insua, D. and Ruggeri, F. (1999) Bayesian forecasting for accident proneness evaluation. Scand. Actuarial J., 134–156.Google Scholar
Ríos Insua, D. and Ruggeri, F. (eds.) (2000) Robust Bayesian analysis. Lecture Notes in Statistics 152, Springer-Verlag, New York.CrossRefGoogle Scholar
Ríos Insua, D., Ruggeri, F. and Vidakovic, B. (1995) Some results on posterior regret Г-minimax estimation. Statistics and Decisions 13, 315–331.Google Scholar
Sivaganesan, S. (1988) Range of posterior measures for priors with arbitrary contaminations. Commun. Statist. – Theory Meth. 17, 1591–1612.CrossRefGoogle Scholar
Sivaganesan, S. and Berger, J.O. (1989) Ranges of posterior measures for priors with unimodal contaminations, Annals of Statist. 17, 868–889.CrossRefGoogle Scholar
Varian, H.R. (1975) A Bayesian approach to real estate assessment. In Studies in Bayesian Econometrics and Statistics, Fienberg, S.E. & Zellner, A. (eds.), North Holland, 195–208.Google Scholar
Vidakovic, B. (2000) Г-minimax: a paradigm for conservative robust Bayesians, in Robust Bayesian Analysis, Insua, D.R. and Ruggeri, F. (eds.), Lectures Notes in Statistics, Springer-Verlag, New York, 241–259.CrossRefGoogle Scholar
Zellner, A. (1986) Bayesian estimation and prediction using asymmetric loss functions. J. Amer. Statist. Assoc. 81, 446–451.CrossRefGoogle Scholar
Zen, M. and Dasgupta, A. (1993) Estimating a binomial parameter: is robust Bayes real Bayes? Statist. & Decisions 11, 37–60.Google Scholar