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On the use of posterior regret \(\Gamma\)-minimax actions to obtain credibility premiums. (English) Zbl 1097.62114

Summary: Computing premiums in a Bayesian context requires the use of a prior distribution that the unknown risk parameter follows in the heterogeneous portfolio. Following the methodology that an actuary only has vague information about this parameter and therefore is unable to specify a simple prior, we choose a class \(\Gamma\) of priors and compute posterior regret \(\Gamma\)-minimax premiums which can be written, under appropriate likelihoods and priors, as a credibility formula.

MSC:

62C20 Minimax procedures in statistical decision theory
91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62F15 Bayesian inference
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[1] Berger, J., Statistical Decision Theory and Bayesian Analysis (1985), Springer-Verlag: Springer-Verlag New York
[2] Berger, J., An overview of robust Bayesian analysis (with discussion), Test, 3, 5-125 (1994) · Zbl 0827.62026
[3] DuMouchel, H., Olshen, A., 1974. On the distributions of claims costs. Statistics Department Technical Report. University of Michigan, EE.UU; DuMouchel, H., Olshen, A., 1974. On the distributions of claims costs. Statistics Department Technical Report. University of Michigan, EE.UU · Zbl 0347.90007
[4] Eichenauer, J.; Lehn, J.; Rettig, S., A gamma-minimax result in credibility theory, Insurance: Mathematics and Economics, 7, 1, 49-57 (1988) · Zbl 0639.62089
[5] Gómez, E.; Hernández, A.; Vázquez-Polo, F. J., Robust Bayesian premium principles in actuarial science, Journal of the Royal Statistical Society, Series D, 49, 2, 241-252 (2000)
[6] Gómez, E.; Pérez, J. M.; Hernández, A.; Vázquez-Polo, F. J., Measuring sensitivity in a bonus-malus system, Insurance: Mathematics and Economics, 31, 1, 105-113 (2002) · Zbl 1037.62110
[7] Heilmann, W., Decision theoretic foundations of credibility theory, Insurance: Mathematics and Economics, 8, 1, 77-95 (1989) · Zbl 0687.62087
[8] Herzog, T. N., Introduction to Credibility Theory (1996), ACTEX Publications: ACTEX Publications Winsted
[9] Hogg, R.; Klugman, S., Loss Distributions (1984), John Wiley & Sons: John Wiley & Sons New York
[10] Klugman, S. A., Bayesian Statistics in Actuarial Science: With Emphasis in Credibility (1992), Kluwer: Kluwer Boston · Zbl 0753.62075
[11] Makov, U. E., Loss robustness via Fisher-weighted squared-error loss function, Insurance: Mathematics and Economics, 16, 1, 1-6 (1995) · Zbl 0836.62006
[12] Ríos, S.; Martín, J.; Ríos, D.; Ruggeri, F., Bayesian forecasting for accident proneness evaluation, Scandinavian Actuarial Journal, 2, 134-156 (1999) · Zbl 0952.91040
[13] Ríos, D.; Ruggeri, F., (Robust Bayesian Analysis. Robust Bayesian Analysis, Lecture Notes in Statistics (2000), Springer: Springer New York) · Zbl 0958.00015
[14] Ríos, D.; Ruggeri, F.; Vidakovic, B., Some results on posterior regret \(\Gamma \)-minimax estimation, Statistics & Decisions, 13, 315-331 (1995) · Zbl 0843.62008
[15] Sarabia, J. M.; Castillo, E.; Gómez, E.; Vázquez-Polo, F., A class of conjugate priors for log-normal claims based on conditional specification, Journal of Risk and Insurance, 72, 3, 479-495 (2004)
[16] Vidakovic, B., \( \Gamma \)-Minimax: A paradigm for conservative robust Bayesians, (Ríos Insua, D.; Ruggeri, F., Robust Bayesian Analysis (2000), Springer-Verlag: Springer-Verlag New York) · Zbl 1281.62040
[17] Watson, S., On Bayesian inference with incompletely specified prior distributions, Biometrika, 61, 1, 193-196 (1974) · Zbl 0275.62003
[18] Zen, M. M.; DasGupta, A., Estimating a binomial parameter: Is robust Bayes real Bayes?, Statistics & Decisions, 11, 37-60 (1993) · Zbl 0767.62003
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