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TESTING FOR RANDOM EFFECTS IN COMPOUND RISK MODELS VIA BREGMAN DIVERGENCE

Published online by Cambridge University Press:  02 July 2020

Himchan Jeong Open the ORCID record for Himchan Jeong [Opens in a new window]
Himchan Jeong*
Affiliation:
Department of Statistics & Actuarial Science Simon Fraser University Burnaby, BCV5A 1S6, Canada E-Mail: [email protected]
*
E-Mail: [email protected]
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Abstract

The generalized linear model (GLM) is a statistical model which has been widely used in actuarial practices, especially for insurance ratemaking. Due to the inherent longitudinality of property and casualty insurance claim datasets, there have been some trials of incorporating unobserved heterogeneity of each policyholder from the repeated observations. To achieve this goal, random effects models have been proposed, but theoretical discussions of the methods to test the presence of random effects in GLM framework are still scarce. In this article, the concept of Bregman divergence is explored, which has some good properties for statistical modeling and can be connected to diverse model selection diagnostics as in Goh and Dey [(2014) Journal of Multivariate Analysis, 124, 371–383]. We can apply model diagnostics derived from the Bregman divergence for testing robustness of a chosen prior by the modeler to possible misspecification of prior distribution both on the naive model, which assumes that random effects follow a point mass distribution as its prior distribution, and the proposed model, which assumes a continuous prior density of random effects. This approach provides insurance companies a concrete framework for testing the presence of nonconstant random effects in both claim frequency and severity and furthermore appropriate hierarchical model which can explain both observed and unobserved heterogeneity of the policyholders for insurance ratemaking. Both models are calibrated using a claim dataset from the Wisconsin Local Government Property Insurance Fund which includes both observed claim counts and amounts from a portfolio of policyholders.

Keywords

Two-part frequency-severity modelcompound random effects modelBregman divergenceBayesian sensitivity analysismodel selectionprior elicitationgeneralized linear model (GLM)generalized Pareto (GP)C11C23C52
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 3 , September 2020 , pp. 777 - 798
Copyright
© 2020 by Astin Bulletin. All rights reserved

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References

Andrews, D.W. (2001) Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica, 69(3), 683–734.CrossRefGoogle Scholar
Berger, J. and Berliner, L.M. (1986) Robust Bayes and empirical Bayes analysis with ϵ-contaminated priors. The Annals of Statistics, 14(2), 461–486.CrossRefGoogle Scholar
Berger, J.O. (1985) Statistical Decision Theory and Bayesian Analysis. Berlin/Heidelberg, Germany: Springer Science & Business Media.CrossRefGoogle Scholar
Berger, J.O., Moreno, E., Pericchi, L.R., Bayarri, M.J., Bernardo, J.M., Cano, J.A., De la Horra, J., Martín, J., Ros-Insúa, D., Betrò, B., Dasgupta, A., Gustafson, P., Wasserman, L., Kadane, J.B., Srinivasan, C., Lavine, M., O’hagen, A., Polasek, W., Robert, C.P., Goutis, C., Ruggeri, F., Salinetti, G. and Sivaganesan, S. (1994) An overview of robust Bayesian analysis. Test, 3, 5–124.CrossRefGoogle Scholar
Boratyńska, A. (2017) Robust Bayesian estimation and prediction of reserves in exponential model with quadratic variance function. Insurance: Mathematics and Economics, 76, 135–140.Google Scholar
Boucher, J.-P., Denuit, M. and Guillén, M. (2008) Models of insurance claim counts with time dependence based on generalization of Poisson and negative binomial distributions. Variance, 2(1), 135–162.Google Scholar
Bregman, L.M. (1967) The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics, 7(3), 200–217.CrossRefGoogle Scholar
Cameron, A.C. and Trivedi, P.K. (2013) Regression Analysis of Count Data, Vol. 53. Cambridge, England: Cambridge University Press.CrossRefGoogle Scholar
Ding, J. and Wang, J.-L. (2008) Modeling longitudinal data with nonparametric multiplicative random effects jointly with survival data. Biometrics, 64(2), 546–556.CrossRefGoogle ScholarPubMed
Dubitzky, W., Wolkenhauer, O., Yokota, H. and Cho, K.-H. (2013) Encyclopedia of Systems Biology. Berlin/Heidelberg, Germany: Springer.CrossRefGoogle Scholar
Eguchi, S. and Kano, Y. (2001) Robustifying maximum likelihood estimation. Technical Report, Institute of Statistical Mathematics.Google Scholar
Eichenauer, J., Lehn, J. and Rettig, S. (1988) A gamma-minimax result in credibility theory. Insurance: Mathematics and Economics, 7(1), 49–57.Google Scholar
Frangos, N.E. and Vrontos, S.D. (2001) Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin: The Journal of the IAA, 31(1), 1–22.CrossRefGoogle Scholar
Frees, E.W., Lee, G. and Yang, L. (2016) Multivariate frequency-severity regression models in insurance. Risks, 4(1), 4.CrossRefGoogle Scholar
Garrido, J., Genest, C. and Schulz, J. (2016) Generalized linear models for dependent frequency and severity of insurance claims. Insurance: Mathematics and Economics, 70, 205–215.Google Scholar
Gelfand, A. and Dey, D. (1991) On Bayesian robustness of contaminated classes of priors. Statistics & Risk Modeling, 9(1–2), 63–80.CrossRefGoogle Scholar
Goh, G. and Dey, D.K. (2014) Bayesian model diagnostics using functional Bregman divergence. Journal of Multivariate Analysis, 124, 371–383.CrossRefGoogle Scholar
Gómez-Déniz, E., Bermúdez, L. and Morillo, I. (2005) Computing bonus–malus premiums under partial prior information. British Actuarial Journal, 11(2), 361–374.CrossRefGoogle Scholar
Gómez-Déniz, E., Hernández, A., Pérez, J. and Vázquez-Polo, F. (2002a) Measuring sensitivity in a bonus–malus system. Insurance: Mathematics and Economics, 31(1), 105–113.Google Scholar
Gómez-Déniz, E., Hernández-Bastida, A. and Vázquez-Polo, F.J. (1999) The Esscher premium principle in risk theory: A Bayesian sensitivity study. Insurance: Mathematics and Economics, 25(3), 387–395.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. Scandinavian Actuarial Journal, 2002(1), 37–44.CrossRefGoogle Scholar
Gómez-Déniz, E., Perez-Sanchez, J.M. and Vázquez-Polo, F.J. (2006) On the use of posterior regret γ-minimax actions to obtain credibility premiums. Insurance: Mathematics and Economics, 39(1), 115–121.Google Scholar
Gómez-Déniz, E. and Vázquez-Polo, F.J. (2005) Modelling uncertainty in insurance bonus–malus premium principles by using a Bayesian robustness approach. Journal of Applied Statistics, 32(7), 771–784.CrossRefGoogle Scholar
Gómez-Déniz, E., Vázquez Polo, F.J. and Bastida, A.H. (2000) Robust Bayesian premium principles in actuarial science. Journal of the Royal Statistical Society: Series D (The Statistician), 49(2), 241–252.CrossRefGoogle Scholar
Insua, S.R., Martin, J., Insua, D.R. and Ruggeri, F. (1999) Bayesian forecasting for accident proneness evaluation. Scandinavian Actuarial Journal, 1999(2), 134–156.CrossRefGoogle Scholar
Jeffreys, H. (1946) An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 186(1007), 453–461.Google Scholar
Jeong, H., Ahn, J., Park, S. and Valdez, E.A. (2020) Generalized linear mixed models for dependent compound risk models. Variance, accepted for publication.Google Scholar
Karimnezhad, A. and Parsian, A. (2018) Bayesian and robust Bayesian analysis in a general setting. Communications in Statistics-Theory and Methods, 48(15), 1–29.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2012) Loss Models: From Data to Decisions, Vol. 715. Hoboken, NJ: John Wiley & Sons.Google Scholar
Lemaire, J. (1998) Bonus-malus systems: The European and Asian approach to merit-rating. North American Actuarial Journal, 2(1), 26–38.CrossRefGoogle Scholar
Ng, E. and Cook, R. (2000) A comparison of some random effect models for parameter estimation in recurrent events. Mathematical and Computer Modelling, 32(1–2), 11–26.CrossRefGoogle Scholar
Peng, F. and Dey, D.K. (1995) Bayesian analysis of outlier problems using divergence measures. Canadian Journal of Statistics, 23(2), 199–213.CrossRefGoogle Scholar
Sánchez-Sánchez, M., Sordo, M., Suárez-Llorens, A. and Gómez-Déniz, E. (2019) Deriving robust Bayesian premiums under bands of prior distributions with applications. ASTIN Bulletin: The Journal of the IAA, 49(1), 147–168.CrossRefGoogle Scholar
Young, V.R. (1998) Robust Bayesian credibility using semiparametric models. ASTIN Bulletin: The Journal of the IAA, 28(2), 187–203.CrossRefGoogle Scholar
Zhang, T. (2004) Statistical behavior and consistency of classification methods based on convex risk minimization. Annals of Statistics, 32(1), 56–85.CrossRefGoogle Scholar
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