Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-19T22:59:38.271Z Has data issue: false hasContentIssue false
  • English
  • Français

BAYESIAN ASYMMETRIC LOGIT MODEL FOR DETECTING RISK FACTORS IN MOTOR RATEMAKING

Published online by Cambridge University Press:  10 January 2014

J.M. Pérez-Sánchez ,
M.A. Negrín-Hernández ,
C. García-García  and
E. Gómez-Déniz
J.M. Pérez-Sánchez*
Affiliation:
Department of Quantitative Methods in Economics, University of Granada, 18011–Granada, Spain
M.A. Negrín-Hernández
Affiliation:
Department of Quantitative Methods, University of Las Palmas de Gran Canaria, 35017–Las Palmas de G.C., Spain E-mail: [email protected]
C. García-García
Affiliation:
Department of Quantitative Methods in Economics, University of Granada, 18011–Granada, Spain E-mail: [email protected]
E. Gómez-Déniz
Affiliation:
Department of Quantitative Methods, University of Las Palmas de Gran Canaria, 35017–Las Palmas de G.C., Spain E-mail: [email protected]
*
E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Modelling automobile insurance claims is a crucial component in the ratemaking procedure. This paper focuses on the probability that a policyholder reports a claim, where the classical logit link does not provide a right model. This is so because databases related with automobile claims are often unbalanced, containing more non-claims than the presence of claims. In this work an asymmetric logit model, which takes into account the large number of non-claims in the portfolio, is considered. Both, logit and asymmetric logit models from a Bayesian point of view, are used to a sample that was collected from a major automobile insurance company in Spain in 2009, resulting in a dataset of 2,000 passenger vehicle. We establish the validity of the asymmetric model in front of the conventional logit link. The use of a garage, the age of the vehicle and the duration of the client's relation with the company are all shown to be significant explanatory variables by the logit model. The asymmetric model includes, in addition, the length of time the policyholder has held a driving licence and the type of use made of the vehicle. The asymmetric model provides a better fit to the data examined.

Keywords

Automobile insurancelogit modelBayesian inferencesymmetric linkasymmetric link
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 44 , Issue 2 , May 2014 , pp. 445 - 457
Copyright
Copyright © ASTIN Bulletin 2014 

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

Albert, J.H. and Chib, S. (1993) Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88 (422), 669–679.CrossRefGoogle Scholar
Albert, J.H. and Chib, S. (1995) Bayesian residual analysis for binary response regression models. Biometrika, 82, 747–769.CrossRefGoogle Scholar
Albrecht, P. (1983) Parametric multiple regression risk models: Connections with tarification, especially in motor insurance. Insurance: Mathematics and Economics, 2, 113–117.Google Scholar
Amemiya, T. (1981) Qualitative response models: A survey. Journal of Economic Literature, 19, 1483–1536.Google Scholar
Basu, S. and Mukhopadhayay, S. (2000) Binary response regression with normal scale mixture links. In Generalized Linear Models: A Bayesian Perspective (eds. Dey, D.K., Ghosh, S.K. and Mallick, B.K.), pp. 231–242. New York: Marcel Dekker.Google Scholar
Bazán, J.L., Branco, M.D. and Bolfarinez, H. (2006) A skew item response model. Bayesian Analysis, 1 (4), 861–892.CrossRefGoogle Scholar
Beirlant, J., Derveaux, V., De Meyer, A.M., Goovaerts, M.J., Labie, E. and Maenhoudt, B. (1991) Statistical risk evaluation applied to (Belgian) car insurance. Insurance: Mathematics and Economics, 10, 289–302.Google Scholar
Bermúdez, L.L. (2009) A priori ratemaking using bivariate Poisson regression models. Insurance: Mathematics and Economics, 44, 135–141.Google Scholar
Bermúdez, L.L., Pérez, J.M., Ayuso, M., Gómez, E. and Vázquez, F.J. (2008) A Bayesian dichotomous model with asymmetric link for fraud in insurance. Insurance: Mathematics and Economics, 42, 779–786.Google Scholar
Besson, J.L. and Partrat, C. (1990) Loi de Poisson inverse Gaussienne et systèmes de bonus–malus. Proceedings of the Astin Colloquium, Montreux, 81, 418–419.Google Scholar
Bolancé, C., Guillén, M. and Pinquet, J. (2003) Time-varying credibility for frequency risk models: Estimation and tests for autoregressive specification on the random effect. Insurance: Mathematics and Economics, 33 (2), 273–282.Google Scholar
Boucher, J.P. and Denuit, M. (2006) Fixed versus random effects in Poisson regression models for claim counts: A case study with motor insurance. ASTIN Bulletin, 36, 285–301.CrossRefGoogle Scholar
Boucher, J.P., Denuit, M. and Guillén, M. (2007) Risk classification for claims counts: A comparative analysis of various zero–inflated mixed Poisson and Hurdle models. North American Actuarial Journal, 11 (4), 110–131.CrossRefGoogle Scholar
Brouhns, N., Denuit, M., Guillén, M. and Pinquet, J. (2003) Bonus–malus scales in segmented tariffs with stochastic migration between segments. Journal of Risk and Insurance, 70, 577–599.CrossRefGoogle Scholar
Carlin, B.P. and Polson, N.G. (1992) Monte Carlo Bayesian methods for discrete regression models and categorical time series. Bayesian Statistics, 4, 577–586.Google Scholar
Chen, M.H., Dey, D.K. and Shao, Q.M. (1999) A new skewed link model for dichotomous quantal response data. Journal of the American Statistical Association, 94, 1172–1186.CrossRefGoogle Scholar
Chen, W.S., Bakshi, B.R., Goel, P.K. and Ungarala, S. (2004) Bayesian estimation of unconstrained nonlinear dynamic systems via sequential Monte Carlo sampling. Industrial & Engineering Chemistry Research, 43 (14), 4012–4025.CrossRefGoogle Scholar
Dean, C.B., Lawless, J.F. and Willmot, G.E. (1989) A mixed Poisson–inverse–Gaussian regression model. Canadian Journal of Statistics, 17, 171–182.CrossRefGoogle Scholar
Denuit, M. (1997) A new distribution of Poisson-type for the number of claims. ASTIN Bulletin, 27, 229–242.CrossRefGoogle Scholar
Denuit, M., Maréchal, X., Pitrebois, S. and Walhin, J.F. (2007) Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus–Malus Systems. Chichester: John Wiley.CrossRefGoogle Scholar
Dierckx, G. (2004) Logistic regression model. In Encyclopedia of Actuarial Science, vol. 1 (eds. Teugels, J. and Sundt, B.). New York: Wiley.Google Scholar
Dionne, G. and Vanasse, C. (1989) A generalization of actuarial automobile insurance rating models: The Negative Binomial distribution with a regression component. ASTIN Bulletin, 19, 199–212.CrossRefGoogle Scholar
Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. (1995) Introducing Markov chain Monte Carlo. In Markov Chain Monte Carlo in Practice (eds. Gilks, W.R., Richardson, S. and Spiegelhalter, D.J.), pp. 1–20. London: Chapman & Hall.CrossRefGoogle Scholar
Gossiaux, A.-M. and Lemaire, J. (1981) Méthodes d'ajustement de distributions de sinistres. Bulletin of the Swiss Association of Actuaries, 81, 87–95.Google Scholar
Guillén, M. (2004) Fraud in insurance. In Encyclopedia of Actuarial Science, vol. 2 (eds. Teugels, J. and Sundt, B.), pp. 729–739. Chichester: John Wiley.Google Scholar
Hausman, J. and McFadden, D. (1984) Specification tests for the multinomial logit model. Econometrica, 52 (5), 1219–1240.CrossRefGoogle Scholar
Holmes, L. and Held, L. (2006) Bayesian auxiliary variables models for binary and multinomial regression. Bayesian Analysis, 1 (1), 145–168.Google Scholar
Kestemont, R.-M. and Paris, J. (1985) Sur l'ajustement du nombre de sinistres. Bulletin of the Swiss Association of Actuaries, 85, 157–163.Google Scholar
Lee, A.H., Stevenson, M.R., Wang, K. and Yau, K.K.W. (2002) Modeling young driver motor vehicle crashes: Data with extra zeros. Accident Analysis and Prevention, 34, 515–521.CrossRefGoogle ScholarPubMed
Lefèvre, Cl. and Picard, Ph. (1996) On the first crossing of a Poisson process in a lower boundary. In Athens Conference on Applied Probability and Time Series, Vol. I, Applied Probability (eds. Heyde, C.C., Prohorov, Yu V., Pyke, R. and Rachev, S.T.), pp. 159–175. Lecture Notes in Statistics. Berlin: Springer, 114.Google Scholar
Lunn, D.J., Thomas, A., Best, N. and Spiegelhalter, D. (2000) WinBUGS: A Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing, 10, 325–337.CrossRefGoogle Scholar
McCulloch, R., Polson, N. and Rossi, P. (1999) A Bayesian analysis of the multinomial probit model with fully identified parameters. Journal of Econometrics, 99 (1), 173–193.CrossRefGoogle Scholar
McFadden, D. (1981) Econometric models of probabilistic choice. In Structural Analysis of Discrete Data (eds. Manski, C. and McFadden, D.), pp. 198–272. Cambridge, MA: MIT Press.Google Scholar
Menard, S.W. (2002) Applied Logistic Regression Analysis, 2nd ed.Thousand Oaks, CA: SAGE.CrossRefGoogle Scholar
O'Hagan, A. (1990) Outliers and credence for location parameter inference. Journal of the American Statistical Association, 85, 172–176.CrossRefGoogle Scholar
Ordaz, J.A. and Melgar, M.C. (2010) Covariate-based pricing of automobile insurance. Insurance Markets and Companies: Analysis and Actuarial Computations, 1 (2), 92–99.Google Scholar
Pinquet, J., Guillén, M. and Bolancé, C. (2001) Allowance for the age of claims in bonus–malus systems. ASTIN Bulletin, 31 (2), 337–348.CrossRefGoogle Scholar
Richaudeau, D. (1999) Automobile insurance contracts and risk of accident: An empirical test using French individual data. Geneva Papers on Risk and Insurance Theory, 24, 97–114.CrossRefGoogle Scholar
Ruohonen, M. (1987) On a model for the claim number process. ASTIN Bulletin, 18, 57–68.CrossRefGoogle Scholar
Sáez-Castillo, A.J., Olmo-Jiménez, M.J., Pérez, J.M., Negrín, M., Arcos, A. and Díaz, J. (2010) Bayesian analysis of nosocomial infection risk and length of stay in a department of general and digestive surgery. Value in Health, 13 (4), 431–439.CrossRefGoogle Scholar
Shankar, V., Milton, F. and Mannering, F. (1997) Modeling accident frequencies as zero-altered probability processes: An empirical inquiry. Accident Analysis and Prevention, 29 (6), 829–837.CrossRefGoogle ScholarPubMed
Spiegelhalter, D.J., Best, N.G., Carlin, B.P. and Van der Linde, A. (2002) Bayesian measures of model complexity and fit (with discussion). Journal of the Royal Statistical Society, Series B, 64 (4), 583–616.CrossRefGoogle Scholar
Stukel, T. (1988) Generalized logistic model. Journal of the American Statistical Association, 83, 426–431.CrossRefGoogle Scholar
Tremblay, L. (1992) Using the Poisson–inverse Gaussian distribution in bonus–malus systems. ASTIN Bulletin, 22, 97–106.CrossRefGoogle Scholar
Willmot, G.E. (1987) The Poisson–inverse Gaussian distribution as an alternative to the negative binomial. Scandinavian Actuarial Journal, 3–4, 113–127.CrossRefGoogle Scholar