ON COMPARISON OF MODELS FOR COUNT DATA WITH EXCESSIVE ZEROS IN NON-LIFE INSURANCE

Abstract

Modeling of the claim frequency is crucial from many respects in the issues of non-life insurance such ratemaking, credibility theory, claim reserving, risk theory, risk classification and bonus-malus system. For analysing claims in non-life insurance the most used models are generalized linear models, depending on the distribution of claims. The distribution of the claim frequency is generally assumed Poisson, however insurance claim data contains zero counts which effects the statistical estimations. In the presence of excess zero, there are more appropriate distributions for the claim frequency such as zero-inflated and hurdle models instead of a standard Poisson distribution. In this study, using a real annual comprehensive insurance data, the zero-inflated claim frequency is modeled via several models with and without consideration of zero-inflation. The underlying models are compared using information criteria and Vuong test. Parameter estimations are carried out using the maximum likelihood.

Keywords

• Claim frequency
• count data
• generalized linear model
• hurdle model
• zero-inflation

References

1. [1] Charpentier, A. (Ed.). (2014) Computational actuarial science with R, CRC press, Chapman and Hall.
2. [2] David, M. (2015) Auto insurance premium calculation using generalized linear models, Procedia Economics and Finance, 20, 147-156.
3. [3] Gschlöβl, S. and Czado, C. (2007) Spatial modelling of claim frequency and claim size in non-life insurance, Scandinavian Actuarial Journal, 2007(3), 202-225.
4. [4] 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.
5. [5] Tse, Y. K. (2009) Nonlife actuarial models: theory, methods and evaluation, Cambridge University Press.
6. [6] Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2008) Modern actuarial risk theory: using R, (Vol. 128), Springer Science & Business Media.
7. [7] Denuit, M., Mar´ echal, X., Pitrebois, S. and Walhin, J. F. (2007) Actuarial modelling of claim counts: Risk classification, credibility and bonus-malus systems, John Wiley & Sons.
8. [8] Boucher, J. P., Denuit, M. and Guill ´en, M. (2007) Risk classification for claim counts: a comparative analysis of various zeroinflated mixed Poisson and hurdle models, North American Actuarial Journal, 11(4), 110-131.

9. [9] Karazsia, B. T. and Van Dulmen, M. H. (2008) Regression models for count data: Illustrations using longitudinal predictors of childhood injury, Journal of pediatric psychology, 33(10), 1076-1084.
10. [10] Lambert, D. (1992) Zero-inflated Poisson regression, with an application to defects in manufacturing, Technometrics, 34(1), 1-14.
11. [11] Yip, K. C. and Yau, K. K. (2005) On modeling claim frequency data in general insurance with extra zeros, Insurance: Mathematics and Economics, 36(2), 153-163.
12. [12] Tüzel, S. and Sucu, M. (2012) Zero-inflated discrete models for claim frequencies, Journal of Statisticians: Statistics and Actuarial Sciences, 5(1), 23-31.
13. [13] Mouatassim, Y. and Ezzahid, E. H. (2012) Poisson regression and zero-inflated Poisson regression: application to private health insurance data, European actuarial journal, 2(2), 187-204.
14. [14] Ismail, N. and Zamani, H. (2013) Estimation of claim count data using negative binomial, generalized Poisson, zero-inflated negative binomial and zero-inflated generalized Poisson regression models, In Casualty Actuarial Society E-Forum (Vol. 41, No. 20, pp. 1-28).
15. [15] Covrig, M. and Badea, D. (2017) Some Generalized Linear Models for the Estimation of the Mean Frequency of Claims in Motor Insurance, Economic Computation and Economic Cybernetics Studies and Research, 51(4), 91-107.
16. [16] Altun, E. (2018) A new zero-inflated regression model with application, Journal of Statisticians: Statistics and Actuarial Sciences, 11 (2), 73-80.
17. [17] Altun, E. (2019) A new model for over-dispersed count data: Poisson quasi-Lindley regression model, Mathematical Sciences, 13 (3), 241-247.
18. [18] Yang, J., Li, X. and Liu, G. F. (2012) Analysis of zero-inflated count data from clinical trials with potential dropouts, Statistics in Biopharmaceutical Research, 4(3), 273-283.
19. [19] Saffari, E.S., Adnan, R., and Greene, W., (2012) Hurdle negative binomial regression model with right cencored count data, Statistics and Operations Research Transactions (SORT), 36(2), pp.181-194.
20. [20] Sarul, L. S. and Sahin, S. (2015) An application of claim frequency data using zero inflated and hurdle models in general insurance, Journal of Business Economics and Finance, 4(4).
21. [21] Gilenko, E. V. and Mironova, E. A. (2017) Modern claim frequency and claim severity models: An application to the Russian motor own damage insurance market, Cogent Economics and Finance, 5(1), 1311097.
22. [22] Baetschmann, G. and Winkelmann, R. (2017) A dynamic hurdle model for zero-inflated count data, Communications in Statistics-Theory and Methods, 46(14), 7174-7187.
23. [23] Sakthivel, K. M. and Rajitha, C. S. (2017) A Comparative Study of Zero-inflated, Hurdle Models with Artificial Neural Network in Claim Count Modeling, International Journal of Statistics and Systems, 12(2), 265-276.
24. [24] McCullagh, P. and Nelder, J. A. (1989) Generalised linear modelling, Chapman and Hall: New York.
25. [25] De Jong, P. and Heller, G.Z., 2008. Generalized Linear Models for Insurance Data, Cambrige: Cambridge University Press.
26. [26] Mullahy, J. (1986). Specification and testing of some modified count data models, Journal of econometrics, 33(3), 341-365.
27. [27] Cameron, C. and Trivedi, P. (1998) Models for count data, Regression Analysis of Count Data, Cambridge University Press, New York.
28. [28] Zeileis, A., Kleiber, C. and Jackman, S. (2008) Regression models for count data in R, Journal of statistical software, 27(8), 1-25.
29. [29] Vuong, Q. H. (1989) Likelihood ratio tests for model selection and non-nested hypotheses, Econometrica: Journal of the Econometric Society, 307-333.
30. [30] Van den Broek, J. (1995) A Score Test for Zero Inflation in a Poisson Distribution. Biometrics, 51(2), 738-743.
31. [31] Cameron, A.C. and Trivedi, P.K. (1990) Regression-based Tests for Overdispersion in the Poisson Model. Journal of Econometrics, 46, 347–364.