Bayesian credibility for GLMs. (English) Zbl 1406.62127
Summary: We revisit the classical credibility results of W. S. Jewell [“Credible means are exact Bayesian for exponential families”, Astin Bull. 8, No. 1, 77–90 (1974; doi:10.1017/S051503610000193)] and H. Bühlmann [“Experience rating and credibility”, ibid. 4, No. 3, 99–207 (1967; doi:10.1017/S0515036100008989)] to obtain credibility premiums for a GLM using a modern Bayesian approach. Here the prior distribution can be chosen without restrictions to be conjugate to the response distribution. It can even come from out-of-sample information if the actuary prefers.
Then we use the relative entropy between the “true” and the estimated models as a loss function, without restricting credibility premiums to be linear. A numerical illustration on real data shows the feasibility of the approach, now that computing power is cheap, and simulations software readily available.
Then we use the relative entropy between the “true” and the estimated models as a loss function, without restricting credibility premiums to be linear. A numerical illustration on real data shows the feasibility of the approach, now that computing power is cheap, and simulations software readily available.
MSC:
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 62F15 | Bayesian inference |
| 91B30 | Risk theory, insurance (MSC2010) |
References:
| [1] | Bernardo, José M., Intrinsic credible regions: an objective Bayesian approach to interval estimation, TEST, 14, 2, 317-384, (2005) · Zbl 1087.62036 |
| [2] | Bühlmann, H., Experience rating and credibility, Astin Bull., 4, 3, 99-207, (1967) |
| [3] | Daniels, H. E., Exact saddlepoint approximations, Biometrika, 67, 1, 59-63, (1980), URL http://biomet.oxfordjournals.org/content/67/1/59.abstract · Zbl 0423.62018 |
| [4] | de Jong, P.; Heller, G. Z., Generalized linear models for insurance data, (2008), Cambridge University Press, URL http://dx.doi.org/10.1017/CBO9780511755408, Cambridge Books Online · Zbl 1142.91046 |
| [5] | De Vylder, F. E., Non-linear regression in credibility theory, Insurance Math. Econom., 4, 3, 163-172, (1985), URL http://EconPapers.repec.org/RePEc:eee:insuma:v:4:y:1985:i:3:p:163-172 · Zbl 0579.62091 |
| [6] | Diaconis, P.; Ylvisaker, D., Conjugate priors for exponential families, Ann. Statist., 7, 2, 269-281, (1979), URL http://dx.doi.org/10.1214/aos/1176344611 · Zbl 0405.62011 |
| [7] | Hachemeister, C.A., 1975. Credibility for regression models with application to trend. In: Proc. of the Berkeley Actuarial Research Conference on Credibility, pp. 129-163.; Hachemeister, C.A., 1975. Credibility for regression models with application to trend. In: Proc. of the Berkeley Actuarial Research Conference on Credibility, pp. 129-163. · Zbl 0354.62057 |
| [8] | Jewell, W. S., Credible means are exact Bayesian for exponential families, Astin Bull., 8, 1, 77-90, (1974) |
| [9] | Johnson, N. L., Uniqueness of a result in the theory of accident proneness, Biometrika, 44, 530-531, (1957) · Zbl 0078.33705 |
| [10] | Jørgensen, B., The theory of exponential dispersion models and analysis of deviance, (1992), Instituto de Matemática Pura e Aplicada, (IMPA) Brazil · Zbl 0983.62502 |
| [11] | Jørgensen, B., The theory of dispersion models, (1997), Chapman & Hall London · Zbl 0928.62052 |
| [12] | Kaas, R.; Goovaerts, M.; Dhaene, J.; Denuit, M., Modern actuarial risk theory, (2008), Springer-Verlag Berlin Heidelberg · Zbl 1148.91027 |
| [13] | Kullback, S., Information theory and statistics, (1968), Dover Publications New York · Zbl 0149.37901 |
| [14] | Kullback, S.; Leibler, R. A., On information and sufficiency, Ann. Math. Statist., 22, 1, 79-86, (1951), URL http://dx.doi.org/10.1214/aoms/1177729694 · Zbl 0042.38403 |
| [15] | Landsman, Z. M.; Makov, U. E., Exponential dispersion models and credibility, Scand. Actuar. J., 1998, 1, 89-96, (1998), URL http://dx.doi.org/10.1080/03461238.1998.10413995 · Zbl 1076.62560 |
| [16] | Nelder, J. A.; Verall, R. J., Credibility theory and generalized linear models, Astin Bull., 27, 1, 71-82, (1997) |
| [17] | Nelder, J. A.; Wedderburn, R. W.M., Generalized linear models, J. Roy. Statist. Soc. Ser. A, Gen., 135, 370-384, (1972) |
| [18] | Ohlsson, E., Combining generalized linear models and credibility models in practice, Scand. Actuar. J., 2008, 4, 301-314, (2008) · Zbl 1224.91080 |
| [19] | R Core Team, 2017. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, URL https://www.R-project.org/; R Core Team, 2017. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, URL https://www.R-project.org/ |
| [20] | Stan Development Team, 2016. RStan: the R interface to Stan, URL http://mc-stan.org/; Stan Development Team, 2016. RStan: the R interface to Stan, URL http://mc-stan.org/ |
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