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A CREDIBILITY APPROACH FOR COMBINING LIKELIHOODS OF GENERALIZED LINEAR MODELS

Published online by Cambridge University Press:  24 May 2016

Marcus C. Christiansen  and
Edo Schinzinger
Marcus C. Christiansen
Affiliation:
Maxwell Institute for Mathematical Sciences Edinburgh, & Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh, EH14 4AS, UK E-Mail: [email protected]
Edo Schinzinger*
Affiliation:
Institute of Insurance Science, University of Ulm, 89081 Ulm, Germany
*
E-Mail: [email protected]
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Abstract

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Generalized linear models are a popular tool for the modelling of insurance claims data. Problems arise with the model fitting if little statistical information is available. In case that related statistics are available, statistical inference can be improved with the help of the borrowing-strength principle. We present a credibility approach that combines the maximum likelihood estimators of individual canonical generalized linear models in a meta-analytic way to an improved credibility estimator. We follow the concept of linear empirical Bayes estimation, which reduces the necessary parametric assumptions to a minimum. The concept is illustrated by a simulation study and an application example from mortality modelling.

Keywords

Linear Bayes estimatorcanonical generalized linear modelmeta-analysispseudo likelihood estimatormulti-population mortality modelling
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 46 , Issue 3 , September 2016 , pp. 531 - 569
Copyright
Copyright © Astin Bulletin 2016 

References

Andersen, E.B. (1980) Discrete Statistical Models with Social Science Applications. Amsterdam: North-Holland Publishing Company.Google Scholar
Antonio, K. and Beirlant, J. (2007) Actuarial statistics with generalized linear mixed models. Insurance: Mathematics and Economics, 40 (1), 58–76.Google Scholar
Brazauskas, V., Dornheim, H. and Ratnam, P. (2014) Credibility and regression modeling. In: Predictive Modeling Applications in Actuarial Science (eds. Frees, E.W., Derrig, R.A. and Meyers, G.), Vol. 1, pp. 217–235. Cambridge University Press.CrossRefGoogle Scholar
Bühlmann, H. (1967) Experience rating and credibility. Astin Bulletin, 4 (03), 199–207. Cambridge University Press.CrossRefGoogle Scholar
Bühlmann, H. and Gisler, A. (2005) A Course in Credibility Theory and its Applications. Springer.Google Scholar
Cairns, A.J., Blake, D. and Dowd, K. (2006) A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance, 73 (4), 687–718.CrossRefGoogle Scholar
Currie, I.D., Durban, M. and Eilers, P.H. (2004) Smoothing and forecasting mortality rates. Statistical Modelling, 4 (4), 279–298.CrossRefGoogle Scholar
De Vylder, F. (1985) Non-linear regression in credibility theory. Insurance: Mathematics and Economics, 4 (3), 163–172.Google Scholar
Efron, B. (1996) Empirical bayes methods for combining likelihoods. Journal of the American Statistical Association, 91 (434), 538–550.CrossRefGoogle Scholar
Fahrmeir, L. and Kaufmann, H. (1983) Konsistenz und asymptotische Normalität des Maximum-Likelihood-Schätzers in verallgemeinerten linearen Modellen. Regensburger Diskussionsbeiträge zur Wirtschaftswissenschaft. Regensburg: Univ., Fak. für Wirtschaftswiss.Google Scholar
Fahrmeir, L. and Kaufmann, H. (1985) Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models. The Annals of Statistics, 13 (1), 342–368.CrossRefGoogle Scholar
Hachemeister, C.A. (1975) Credibility for regression models with application to trend. In Credibility, Theory and Applications, Proceedings of the berkeley Actuarial Research Conference on Credibility, pp. 129–163. Berkeley.Google Scholar
Human-Mortality-Database (2014) University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de (data downloaded on December 29th, 2014).Google Scholar
Klenke, A. (2006) Wahrscheinlichkeitstheorie, Vol. 1. Heidelberg: Springer.Google Scholar
Lo, C.H., Fung, W.K. and Zhu, Z.Y. (2007) Structural parameter estimation using generalized estimating equations for regression credibility models. Astin Bulletin, 37 (02), 323–343.CrossRefGoogle Scholar
McFadden, D. (1973) Conditional logit analysis of qualitative choice behavior, pp. 105–142. Institute of Urban and Regional Development, University of California.Google Scholar
Nelder, J. and Verrall, R. (1997) Credibility theory and generalized linear models. Astin Bulletin, 24, 71–82.CrossRefGoogle Scholar
Neuhaus, W. (1985) Choice of statistic in linear Bayes estimation. Scandinavian Actuarial Journal, 1985 (1), 1–26.CrossRefGoogle Scholar
Norberg, R. (1980) Empirical Bayes credibility. Scandinavian Actuarial Journal, 1980 (4), 177–194.CrossRefGoogle Scholar
Norberg, R. (2004) Credibility theory. In Encyclopedia of Actuarial Science (eds. Teugels, J.L. and Sundt, B.) Chichester, UK: Wiley.Google Scholar
Ohlsson, E. (2008) Combining generalized linear models and credibility models in practice. Scandinavian Actuarial Journal, 2008 (4), 301–314.CrossRefGoogle Scholar
Ohlsson, E. and Johansson, B. (2006) Exact credibility and tweedie models. Astin Bulletin, 36 (1), 121.CrossRefGoogle Scholar
Pitselis, G. (2004) De vylder's robust nonlinear regression credibility. Beligan Actuarial Bulletin, 1 (4), 44–49.Google Scholar
Pratt, J.W. (1959) On a general concept of “in probability". The Annals of Mathematical Statistics, 30 (2), 549–558.CrossRefGoogle Scholar
Qian, W. (2000) An application of nonparametric regression estimation in credibility theory. Insurance: Mathematics and Economics, 27 (2), 169–176.Google Scholar
Taylor, G.C. (1977) Abstract credibility. Scandinavian Actuarial Journal, 1977 (3), 149–168.CrossRefGoogle Scholar
Whitney, A. (1918) The theory of experience rating. Proceedings of the Casualty Actuarial Society, (4), 274–292.Google Scholar
Witting, H. and Nölle, G. (1970) Angewandte mathematische Statistik: Optimale finite und asymptotische Verfahren, Vol. 14, Stuttgart: Teubner.Google Scholar