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Credibility using copulas. (English) Zbl 1085.62121

Summary: Credibility is a form of insurance pricing that is widely used, particularly in North America. The theory of credibility has been called a ”cornerstone” in the field of actuarial science. Students of the North American actuarial bodies also study loss distributions, the process of statistical inference of relating a set of data to a theoretical (loss) distribution. In this work, we develop a direct link between credibility and loss distributions through the notion of a copula, a tool for understanding relationships among multivariate outcomes.
This paper develops credibility using a longitudinal data framework. In a longitudinal data framework, one might encounter data from a cross section of risk classes (towns) with a history of insurance claims available for each risk class. For the marginal claims distributions, we use generalized linear models, an extension of linear regression that also encompasses Weibull and Gamma regressions. Copulas are used to model the dependencies over time; specifically, this paper is the first to propose using a t-copula in the context of generalized linear models. The t-copula is the copula associated with the multivariate t-distribution; like the univariate t-distributions, it seems especially suitable for empirical work. Moreover, we show that the t-copula gives rise to easily computable predictive distributions that we use to generate credibility predictors. Like Bayesian methods, our copula credibility prediction methods allow us to provide an entire distribution of predicted claims, not just a point prediction. We present an illustrative example of Massachusetts automobile claims, and compare our new credibility estimates with those currently existing in the literature.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62J12 Generalized linear models (logistic models)
62F15 Bayesian inference
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References:

[1] Bailey Arthur, Proceedings of the Casualty Actuarial Society pp 32– (1945)
[2] Bailey Arthur, Proceedings of the Casualty Actuarial Society 37 pp 7– (1950)
[3] Bühlmann Hans, ASTIN Bulletin 4 pp 199– (1967)
[4] Demarta Stefano, The t-Copula and Related Copulas (2004) · Zbl 1104.62060
[5] Embrechts Paul, Modeling Dependence with Copulas and Applications to Risk Management (2001)
[6] Frees Edward W., North American Actuarial Journal 7 (1) pp 13– (2003) · Zbl 1084.62110
[7] Bailey Arthur, Longitudinal and Panel Data: Analysis and Applications for the Social Sciences (2004)
[8] Frees Edward W., North American Actuarial Journal 2 (1) pp 1– (1998) · Zbl 1081.62564
[9] Frees Edward W., Insurance: Mathematics and Economics 24 pp 229– (1999) · Zbl 0945.62112
[10] Frees Edward W., North American Actuarial Journal 4 (4) pp 24– (2001) · Zbl 1083.91538
[11] Haberman Steven, The Statistician 45 (4) pp 407– (1996)
[12] Hickman James C., North American Actuarial Journal 3 (2) pp 1– (1999) · Zbl 1082.62533
[13] Jewell William S., Astin Bulletin 8 (1) pp 77– (1974)
[14] Joe Harry, Multivariate Models and Dependence Concepts (1997)
[15] Johnson Norman L., Distributions in Statistics: Continuous Multivariate Distributions (1972) · Zbl 0248.62021
[16] Keffer Ralph, Transactions of the Actuarial Society of America 30 pp 130– (1929)
[17] Klugman Stuart A., Bayesian Statistics in Actuarial Science (1992) · Zbl 0753.62075
[18] Klugman Stuart A., Loss Models: From Data to Decisions (1998) · Zbl 0905.62104
[19] Lambert Philippe, Statistics in Medicine 15 pp 1695– (1996)
[20] Lambert Philippe, Statistics in Medicine 21 pp 3197– (2002) · Zbl 1111.62351
[21] Mccullagh P., Generalized Linear Models, 2. ed. (1989) · Zbl 0744.62098
[22] Meester Steven G., Biometrics 50 (4) pp 954– (1994) · Zbl 0825.62788
[23] Miller Robert B., Credibility–Theory and Applications (1975)
[24] Mowbray Albert H., Proceedings of the Casualty Actuarial Society 1 pp 24– (1914)
[25] Nelsen Roger B., An Introduction to Copulas (1999)
[26] Pinquet Jean, ASTIN Bulletin 27 pp 33– (1997)
[27] Venter Gary G., Casualty Actuarial Society Forum pp 215– (2003)
[28] Whitney Albert W., Proceedings of the Casualty Actuarial Society pp 4– (1918)
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