Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T01:55:01.106Z Has data issue: false hasContentIssue false
  • English
  • Français

A CLASS OF MIXTURE OF EXPERTS MODELS FOR GENERAL INSURANCE: APPLICATION TO CORRELATED CLAIM FREQUENCIES

Published online by Cambridge University Press:  04 September 2019

Tsz Chai Fung ,
Andrei L. Badescu  and
X. Sheldon Lin
Tsz Chai Fung
Affiliation:
Department of Statistical Sciences University of Toronto100 St George Street Toronto, ON M5S 3G3, Canada E-Mail: tszchai.fung@mail.utoronto.ca
Andrei L. Badescu
Affiliation:
Department of Statistical Sciences University of Toronto100 St George Street Toronto, ON M5S 3G3, Canada E-Mail: badescu@utstat.toronto.edu
X. Sheldon Lin
Affiliation:
Department of Statistical Sciences University of Toronto100 St George Street Toronto, ON M5S 3G3, Canada E-Mail: sheldon@utstat.toronto.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper focuses on the estimation and application aspects of the Erlang count logit-weighted reduced mixture of experts model (EC-LRMoE), which is a fully flexible multivariate insurance claim frequency regression model. We first prove the identifiability property of the proposed model to ensure that it is a suitable candidate for statistical inference. An expectation conditional maximization (ECM) algorithm is developed for efficient model calibrations. Three simulation studies are performed to examine the effectiveness of the proposed ECM algorithm and the versatility of the proposed model. The applicability of the EC-LRMoE is shown through fitting an European automobile insurance data set. Since the data set contains several complex features, we find it necessary to adopt such a flexible model. Apart from showing excellent fitting results, we are able to interpret the fitted model in an insurance perspective and to visualize the relationship between policyholders’ information and their risk level. Finally, we demonstrate how the fitted model may be useful for insurance ratemaking.

Keywords

Erlang count modelsexpectation conditional maximization algorithmlogit-weighted gating functionsmixture of experts modelsmultivariate count regression
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 3 , September 2019 , pp. 647 - 688
Copyright
© Astin Bulletin 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S., Nerman, O. and Olsson, M. (1996). Fitting phase-type distributions via the EM algorithm. Scandinavian Journal of Statistics 23(4), 419–441.Google Scholar
Badescu, A. L., Lin, X. S., Tang, D. and Valdez, E. A. (2015). Multivariate Pascal mixture regression models for correlated claim frequencies. Available in SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2618265.Google Scholar
Bermúdez, L. (2009). A priori ratemaking using bivariate Poisson regression models. Insurance: Mathematics and Economics 44(1), 135–141.Google Scholar
Bermúdez, L. and Karlis, D. (2011). Bayesian multivariate Poisson models for insurance ratemaking. Insurance: Mathematics and Economics 48(2), 226–236.Google Scholar
Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. Heidelberg: Springer.CrossRefGoogle Scholar
Conway, R. W. and Maxwell, W. L. (1962). A queuing model with state dependent service rates. Journal of Industrial Engineering 12(2), 132–136.Google Scholar
Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 39(1), 1–38.CrossRefGoogle Scholar
Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96(456), 1348–1360.CrossRefGoogle Scholar
Frees, E. W., Lee, G. and Yang, L. (2016). Multivariate frequency/severity regression models in insurance. Risks 4(1), 4.CrossRefGoogle Scholar
Friedman, J. H. (2001). Greedy function approximation: a gradient boosting machine. The Annals of Statistics 29(5), 1189–1232.CrossRefGoogle Scholar
Fung, T. C., Badescu, A. L. and Lin, X. S. (2019a). A class of mixture of experts models for general insurance: Theoretical developments. Submitted.CrossRefGoogle Scholar
Fung, T. C., Badescu, A. L. and Lin, X. S. (2019b). Multivariate Cox hidden Markov models with an application to operational risk. Scandinavian Actuarial Journal, accepted.CrossRefGoogle Scholar
Gui, W., Huang, R. and Lin, X. S. (2018). Fitting the Erlang mixture model to data via a GEM-CMM algorithm. Journal of Computational and Applied Mathematics 343, 189–205.CrossRefGoogle Scholar
Jacobs, R. A., Jordan, M. I., Nowlan, S. J. and Hinton, G. E. (1991). Adaptive mixtures of local experts. Neural Computation 3(1), 79–87.CrossRefGoogle ScholarPubMed
Jiang, W. and Tanner, M. A. (1999). On the identifiability of mixtures-of-experts. Neural Networks 12(9), 1253–1258.CrossRefGoogle ScholarPubMed
Jordan, M. I. and Jacobs, R. A. (1994). Hierarchical mixtures of experts and the EM algorithm. Neural Computation 6(2), 181–214.CrossRefGoogle Scholar
Kuha, J. (2004). AIC and BIC: Comparisons of assumptions and performance. Sociological Methods & Research 33(2), 188–229.CrossRefGoogle Scholar
Lee, S. C. K. and Lin, X. S. (2010). Modeling and evaluating insurance losses via mixtures of Erlang distributions. North American Actuarial Journal 14(1), 107–130.CrossRefGoogle Scholar
Lee, S. C. K. and Lin, X. S. (2012). Modeling dependent risks with multivariate Erlang mixtures. ASTIN Bulletin: The Journal of the IAA 42(1), 153–180.Google Scholar
Lord, D., Guikema, S. D. and Geedipally, S. R. (2008). Application of the Conway–Maxwell–Poisson generalized linear model for analyzing motor vehicle crashes. Accident Analysis & Prevention 40(3), 1123–1134.CrossRefGoogle ScholarPubMed
McLachlan, G. and Peel, D. (2000). Finite Mixture Models. Wiley Series in Probability and Statistics.CrossRefGoogle Scholar
Meng, X.-L. and Rubin, D. B. (1993). Maximum likelihood estimation via the ECM algorithm: A general framework. Biometrika 80(2), 267–278.CrossRefGoogle Scholar
Shi, P. and Valdez, E. A. (2014). Multivariate negative binomial models for insurance claim counts. Insurance: Mathematics and Economics 55, 18–29.Google Scholar
Teicher, H. (1963). Identifiability of finite mixtures. The Annals of Mathematical Statistics, 34(4), 1265–1269.CrossRefGoogle Scholar
Teicher, H. (1967). Identifiability of mixtures of product measures. The Annals of Mathematical Statistics 38(4), 1300–1302.CrossRefGoogle Scholar
Wedel, M. and DeSarbo, W. S. (1995). A mixture likelihood approach for generalized linear models. Journal of Classification 12(1), 21–55.CrossRefGoogle Scholar
Winkelmann, R. (1995). Duration dependence and dispersion in count-data models. Journal of Business & Economic Statistics 13(4), 467–474.Google Scholar
Yin, C. and Lin, X. S. (2016). Efficient estimation of Erlang mixtures using iSCAD penalty with insurance application. ASTIN Bulletin: The Journal of the IAA 46(3), 779–799.CrossRefGoogle Scholar