Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access July 27, 2020

Bayesian credibility premium with GB2 copulas

  • Himchan Jeong and Emiliano A. Valdez EMAIL logo
From the journal Dependence Modeling

Abstract

For observations over a period of time, Bayesian credibility premium may be used to predict the value of a response variable for a subject, given previously observed values. In this article, we formulate Bayesian credibility premium under a change of probability measure within the copula framework. Such reformulation is demonstrated using the multivariate generalized beta of the second kind (GB2) distribution. Within this family of GB2 copulas, we are able to derive explicit form of Bayesian credibility premium. Numerical illustrations show the application of these estimators in determining experience-rated insurance premium. We consider generalized Pareto as a special case.

MSC 2010: 62E15; 62F15; 62P05

References

[1] Bailey, A. L. (1950). Credibility procedures: Laplace’s generalization of Bayes’ rule and the combination of collateral knowledge with observed data. Proceeding of the Casualty Actuarial Society 37(67), pp. 7–23.Search in Google Scholar

[2] Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis. Springer, New York.10.1007/978-1-4757-4286-2Search in Google Scholar

[3] Bühlmann, H. (1967). Experience rating and credibility. Astin Bull. 4(3), 199–207.10.1017/S0515036100008989Search in Google Scholar

[4] Bühlmann, H. and A. Gisler (2005). A Course in Credibility Theory and its Applications. Springer, Berlin.Search in Google Scholar

[5] Frees, E. W. and E. A. Valdez (1998). Understanding relationships using copulas. N. Am. Actuar. J. 2(1), 1–25.10.1080/10920277.1998.10595667Search in Google Scholar

[6] Hougaard, P., B. Harvald, and N. V. Holm (1992). Measuring the similarities between the lifetimes of adult danish twins born between 1881–1930. J. Amer. Statist. Assoc. 87(417), 17–24.Search in Google Scholar

[7] Jewell, W. S. (1974). Credible means are exact bayesian for exponential families. Astin Bull. 8(1), 77–90.10.1017/S0515036100009193Search in Google Scholar

[8] Joe, H. (1997). Multivariate Models and Multivariate Dependence Concepts. Chapman and Hall, London.Search in Google Scholar

[9] Li, D. X. (2000). On default correlation: a copula function approach. J. Fixed Income 9(4), 43–54.10.3905/jfi.2000.319253Search in Google Scholar

[10] Mayerson, A. L. (1964). A bayesian view of credibility. Proceeding of the Casualty Actuarial Society 51(95), pp. 85–104.Search in Google Scholar

[11] McDonald, J. B. and R. M. Bookstaber (1991). Option pricing for generalized distributions. Comm. Statist. Theory Methods 20(12), 4053–4068.10.1080/03610929108830756Search in Google Scholar

[12] McDonald, J. B. and R. J. Butler (1990). Regression models for positive random variables. J. Econometrics 43(1-2), 227–251.10.1016/0304-4076(90)90118-DSearch in Google Scholar

[13] Morris, C. N. (1983). Parametric empirical Bayes inference: theory and applications. J. Amer. Statist. Assoc. 78(381), 47–65.10.1080/01621459.1983.10477920Search in Google Scholar

[14] Salvadori, G. and C. De Michele (2007). On the use of copulas in hydrology: theory and practice. J. Hydrol. Eng. 12(4), 369–380.10.1061/(ASCE)1084-0699(2007)12:4(369)Search in Google Scholar

[15] Shih, J. H. and T. A. Louis (1995). Inferences on the association parameter in copula models for bivariate survival data. Biometrics 51(4), 1384–1399.10.2307/2533269Search in Google Scholar

[16] Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231.Search in Google Scholar

[17] Yang, X., E. W. Frees, and Z. Zhang (2011). A generalized beta copula with applications in modeling multivariate long-tailed data. Insurance Math. Econom. 49(2), 265–284.10.1016/j.insmatheco.2011.04.007Search in Google Scholar

Received: 2020-03-11
Accepted: 2020-06-23
Published Online: 2020-07-27

© 2020 Himchan Jeong et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 4.2.2023 from https://www.degruyter.com/document/doi/10.1515/demo-2020-0009/html
Scroll Up Arrow