On modeling claim frequency data in general insurance with extra zeros. (English) Zbl 1070.62098

Summary: In some occasions, claim frequency data in general insurance may not follow the traditional Poisson distribution and in particular they are zero-inflated. Extra dispersion appears as the number of observed zeros exceeding the number of expected zeros under the Poisson or even the negative binomial distribution assumptions. This paper presents several parametric zero-inflated count distributions, including the ZIP, ZINB, ZIGP and ZIDP, to accommodate the excess zeros for insurance claim count data. Different count distributions in the second component are considered to allow flexibility to control the distribution shape. The generalized Pearson \(\chi^{2}\) statistic, Akaike’s information criteria (AIC) and Bayesian information criteria (BIC) are used as goodness-of-fit and model selection measures. With the presence of extra zeros in a data set of automobile insurance claims, our result shows that the application of zero-inflated count data models and in particular the zero-inflated double Poisson regression model, provide a good fit to the data.


62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Angers, J.F.; Biswas, A., A Bayesian analysis of zero-inflated generalized Poisson model, Comput. stat. data anal., 42, 37-46, (2003) · Zbl 1429.62091
[2] Böhning, D.; Dietz, E.; Schlattmann, P.; Mendonca, L.; Kirchner, U., The zero-inflated Poisson model and the decayed, missing and filled teeth index in dental epidemiology, J. R. stat. soc.: ser. A, 162, 195-209, (1999)
[3] Carriere, J., Nonparametric tests for mixed Poisson distributions, Ins.: mathematics econ., 12, 3-8, (1993) · Zbl 0768.62031
[4] Carrivick, P.J.W.; Lee, A.H.; Yau, K.K.W., Zero-inflated Poisson modeling to evaluate occupational safety interventions, Safety sci., 41, 53-63, (2003)
[5] Dalrymple, M.L.; Hudson, I.L.; Ford, R.P.K., Finite mixture, zero-inflated Poisson and hurdle models with application to SIDS, Comput. stat. data anal., 41, 491-504, (2003) · Zbl 1429.62513
[6] Dobbie, M.J.; Welsh, A.H., Models for zero-inflated count data using the Neyman type A distribution, Stat. modell., 1, 65-80, (2001) · Zbl 0983.62085
[7] Efron, B., Double exponential families and their use in generalized linear regression, J. am. stat. assoc., 81, 709-721, (1986) · Zbl 0611.62072
[8] Gupta, P.L.; Gupta, R.C.; Tripathi, R.C., Analysis of zero-adjusted count data, Comput. stat. data anal., 23, 207-218, (1996) · Zbl 0875.62096
[9] Gurmu, S., Generalized hurdle count data regression models, Econ. lett., 58, 263-268, (1998) · Zbl 0909.90083
[10] Haberman, S.; Renshaw, A.E., Generalized linear models and actuarial science, The Statistician, 45, 407-436, (1996)
[11] Heilbron, D., Zero-altered and other regression models for count data with added zeros, Biometrical J., 36, 531-547, (1994) · Zbl 0846.62053
[12] Hürlimann, W., On maximum likelihood estimation for count data models, Ins.: mathematics econ., 9, 39-49, (1990) · Zbl 0724.62103
[13] Lambert, D., Zero-inflated Poisson regression, with an application to defects in manufacturing, Technometrics, 34, 1-14, (1992) · Zbl 0850.62756
[14] Lee, A.H.; Stevenson, M.R.; Wang, K.; Yau, K.K.W., Modeling Young driver motor vehicle crashes: data with extra zeros, Accid. anal. prev., 34, 515-521, (2002)
[15] Li, C.S.; Lu, J.C.; Park, J.; Kim, K.; Brinkley, P.A.; Peterson, J.P., Multivariate zero-inflated Poisson models and their applications, Technometrics, 41, 29-38, (1999)
[16] McCullagh, P.; Nelder, J.A., Generalized linear models, (1989), Chapman and Hall New York · Zbl 0744.62098
[17] Partrat, C., Compound model for two dependent kinds of claim, Ins.: mathematics econ., 15, 219-231, (1994) · Zbl 0820.62090
[18] Renshaw, A.E., Modelling the claims process in the presence of covariates, Astin bull., 24, 265-285, (1994)
[19] SAS Institute Inc., 1998. Solving Business Problems Using SAS Enterprise Miner Software. SAS Institute White Paper, SAS Institute Inc., Cary, NC.
[20] Shankar, V.; Milton, J.; Mannering, F., Modelling accident frequencies as zero-altered probability process: an empirical inquiry, Accid. anal. prev., 29, 829-837, (1997)
[21] Thomas, H.; Samson, D., Linear models as aids in insurance decision making: the estimation of automobile insurance claims, J. business res., 15, 247-256, (1987)
[22] Van den Broek, J., A score test for zero inflation in a Poisson distribution, Biometrics, 51, 738-743, (1995) · Zbl 0825.62377
[23] Wedderburn, R.W.M., Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method, Biometrika, 61, 439-447, (1974) · Zbl 0292.62050
[24] Welsh, A.H.; Cunningham, R.B.; Donnelly, C.F.; Lindenmayer, D.B., Modelling the abundance of rare species: statistical models for counts with extra zeros, Ecol. modell., 88, 297-308, (1996)
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