Abstract
This article introduces a k-Inflated Negative Binomial mixture distribution/regression model as a more flexible alternative to zero-inflated Poisson distribution/regression model. An EM algorithm has been employed to estimate the model’s parameters. Then, such new model along with a Pareto mixture model have employed to design an optimal rate–making system. Namely, this article employs number/size of reported claims of Iranian third party insurance dataset. Then, it employs the k-Inflated Negative Binomial mixture distribution/regression model as well as other well developed counting models along with a Pareto mixture model to model frequency/severity of reported claims in Iranian third party insurance dataset. Such numerical illustration shows that: (1) the k-Inflated Negative Binomial mixture models provide more fair rate/pure premiums for policyholders under a rate–making system; and (2) in the situation that number of reported claims uniformly distributed in past experience of a policyholder (for instance
Acknowledgements
Authors would like to thank professor Jean-Philippe Boucher and professor Jean Lemaire for their constructive suggestions which improved results and presentation of this article. Thanks to an anonymous reviewer for his/her constructive comments.
References
Alfó, M., and Trovato G. 2004. “Semiparametric Mixture Models for Multivariate Count Data, with Application.” The Econometrics Journal 7 (2): 426–454.10.1111/j.1368-423X.2004.00138.xSearch in Google Scholar
Aryuyuen, S., and Bodhisuwan W. 2013. “The Negative Binomial-Generalized Exponential (NB-GE) Distribution.” Applied Mathematical Sciences 7 (22): 1093–1105.10.12988/ams.2013.13099Search in Google Scholar
Bermúdez, L., and Karlis D. 2011. “Bayesian Multivariate Poisson Models for Insurance Ratemaking.” Insurance: Mathematics and Economics 48 (2): 226–236.10.1016/j.insmatheco.2010.11.001Search in Google Scholar
Boucher, J. P., and Denuit M. 2008. “Credibility Premiums for the Zero-Inflated Poisson Model and New Hunger for Bonus Interpretation.” Insurance: Mathematics and Economics 42 (2): 727–735.10.1016/j.insmatheco.2007.08.003Search in Google Scholar
Boucher, J. P., Denuit M., and Guillén M. 2007. “Risk Classification for Claim Counts: A Comparative Analysis of Various Zero inflated Mixed Poisson and Hurdle Models.”North American Actuarial Journal 11 (4): 110–131.10.1080/10920277.2007.10597487Search in Google Scholar
Boucher, J. P., Denuit M., and Guillén M. 2009. “Number of Accidents or Number of Claims? An Approach with Zero-Inflated Poisson Models for Panel Data.” Journal of Risk and Insurance 76 (4): 821–846.10.1111/j.1539-6975.2009.01321.xSearch in Google Scholar
Boucher, J. P., and Inoussa R. 2014. “A Posteriori Ratemaking with Panel Data.” Astin Bulletin 44 (3): 587–612.10.1017/asb.2014.11Search in Google Scholar
Brännaäs, K., and Rosenqvist G. 1994. “Semiparametric Estimation of Heterogeneous Count Data Models.” European Journal of Operational Research 76 (2): 247–258.10.1016/0377-2217(94)90105-8Search in Google Scholar
Denuit, M., and Dhaene J. 2001. “Bonus–Malus Scales using Exponential Loss Functions.” Blatter der Deutsche Gesellschaft fur Versicherungsmathematik 25 (1): 13–27.10.1007/BF02857113Search in Google Scholar
Denuit, M., Maréchal X., Pitrebois S., and Walhin J. F. 2007. Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus–Malus Systems. New York: John Wiley and Sons.10.1002/9780470517420Search in Google Scholar
Dionne, G., and Vanasse C. 1989. “A Generalization of Automobile Insurance Rating Models: the Negative Binomial Distribution with a Regression Component.Astin Bulletin 19 (2): 199–212.10.2143/AST.19.2.2014909Search in Google Scholar
Dionne, G., and Vanasse C. 1992. “Automobile Insurance Ratemaking in the Presence of Asymmetrical Information.” Journal of Applied Econometrics 7 (2): 149–165.10.1002/jae.3950070204Search in Google Scholar
Frangos, N. E., and Vrontos S. D. 2001. “Design of Optimal Bonus–Malus Systems with a Frequency and a Severity Component on an Individual Basis in Automobile Insurance.” Astin Bulletin 31 (1): 1–22.10.2143/AST.31.1.991Search in Google Scholar
Greene, W. H. 1994. Accounting for Excess Zeros and Sample Selection in Poisson and Negative Binomial Regression Models. Technical Report No. EC-94-10, Department of Economics, Stern School of Business, New York University.Search in Google Scholar
Gómez-Déniz, E., Sarabia J. M., and Calderín-Ojeda E. 2008. “Univariate and Multivariate Versions of the Negative Binomial-Inverse Gaussian Distributions with Applications.” Insurance: Mathematics and Economics 42 (1): 39–49.10.1016/j.insmatheco.2006.12.001Search in Google Scholar
Hall, D. B. 2000. “Zero-Inflated Poisson and Binomial Regression with Random Effects: A Case Study.” Biometrics 56 (4): 1030–1039.10.1111/j.0006-341X.2000.01030.xSearch in Google Scholar
Johnson, N. L., Kemp A. W., and Kotz S. 2005. Univariate Discrete Distributions (Vol. 444). New York: John Wiley and Sons.10.1002/0471715816Search in Google Scholar
Laird, N. 1978. “Nonparametric Maximum Likelihood Estimation of a Mixing Distribution.” Journal of the American Statistical Association 73, 805–811.10.1080/01621459.1978.10480103Search in Google Scholar
Lambert, D. 1992. “Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing.” Technometrics 34 (1): 1–14.10.2307/1269547Search in Google Scholar
Lange, J. T. 1969. “Application of a Mathematical Concept of Risk to Property-Liability Insurance Ratemaking.” Journal of Risk and Insurance 34 (4) 383–391.10.2307/251296Search in Google Scholar
Lemaire, J. 1995. “Bonus-Malus Systems in Automobile Insurance.” Insurance Mathematics and Economics 3 (16): 277.10.1007/978-94-011-0631-3Search in Google Scholar
Lim, H. K., Li W. K., and Philip L. H. 2014. “Zero-Inflated Poisson Regression Mixture Model.” Computational Statistics and Data Analysis 71, 151–158.10.1016/j.csda.2013.06.021Search in Google Scholar
McLachlan, G., and Krishnan, N., 1997. The EM algorithm and Extensions. Wiley, New York.Search in Google Scholar
Meng, S., Yuan W., and Whitmore G. A. 1999. “Accounting for Individual Over-dispersion in a Bonus–Malus Automobile Insurance System.” Astin Bulletin 29, 327–338.10.2143/AST.29.2.504619Search in Google Scholar
Panjer, H. H., and Willmot G. E. 1981. “Finite sum Evaluation of the Negative Binomial-Exponential Model.” Astin Bulletin 12 (2): 133–137.10.1017/S0515036100007066Search in Google Scholar
Payandeh Najafabadi, A. T. 2010. “A New Approach to the Credibility Formula. Insurance: Mathematics and Economics 46 (2): 334–338.10.1016/j.insmatheco.2009.11.007Search in Google Scholar
Payandeh Najafabadi, A. T., Atatalab F., and Omidi Najafabadi M. 2015. “Credibility Premium for Rate–Making Systems.” Communications in Statistics-Theory and Methods (just-accepted).10.1080/03610926.2014.995823Search in Google Scholar
Pinquet, J. 1997. “Allowance for Cost of Claims in Bonus-Malus Systems.” Astin Bulletin 27 (1): 33–57.10.2143/AST.27.1.542066Search in Google Scholar
Pinquet, J. 1998. “Designing Optimal Bonus-Malus Systems from Different Types of Claims.” Astin Bulletin 28 (2): 205–220.10.2143/AST.28.2.519066Search in Google Scholar
Pudprommarat, C. 2012. Negative Binomial-Beta Exponential Distribution. Doctoral dissertation, Kasetsart University.Search in Google Scholar
Pudprommarat, C., Bodhisuwan W., and Zeephongsekul P. 2012. “A New Mixed Negative Binomial Distribution.” Journal of Applied Sciences 12 (17): 1853–1858.10.3923/jas.2012.1853.1858Search in Google Scholar
Rigby, R. A., and Stasinopoulos D. M. 2001 July. “The GAMLSS Project: A Flexible Approach to Statistical Modelling.” In New trends in statistical modelling: Proceedings of the 16th international workshop on statistical modelling, 337–345.Search in Google Scholar
Rigby, R. A., and Stasinopoulos D. M. 2009. “A Flexible Regression Approach Using GAMLSS in R. Lancaster, England.Search in Google Scholar
Simar, L. 1976. “Maximum Likelihood Estimation of a Compound Poisson Process.” The Annals of Statistics 4 (6): 1200–1209.10.1214/aos/1176343651Search in Google Scholar
Simon, L. J. 1961. “Fitting Negative Binomial Distributions by the Method of Maximum Likelihood.” Proceedings of the Casualty Actuarial Society 48: 45–53.Search in Google Scholar
Tzougas, G., and Fragos N. 2013. “Design of an Optimal Bonus-Malus System using the Sichel Distribution as a Model of Claim Counts.” Available at SSRN 2188289.10.2139/ssrn.2188289Search in Google Scholar
Tzougas, G., and Frangos N. 2014a. “The Design of an Optimal Bonus-Malus System Based on the Sichel Distribution.” In Modern Problems in Insurance Mathematics, 239-260. Springer International Publishing.10.1007/978-3-319-06653-0_15Search in Google Scholar
Tzougas, G., and Frangos N. 2014b. “The Design of an Optimal Bonus-Malus System Based on the Sichel Distribution.” In Modern Problems in Mathematics Insurance, 239-260. New York: Springer.10.1007/978-3-319-06653-0_15Search in Google Scholar
Tzougas, G., Vrontos S., and Frangos N. 2014. “Optimal Bonus-Malus Sstems using Finite Mixture Models.” Astin Bulletin 44 (2): 417–444.10.1017/asb.2013.31Search in Google Scholar
Walhin, J. F., and Paris J. 1999. “Using Mixed Poisson Processes in Connection with Bonus-Malus Systems.” Astin Bulletin 29 (1): 81–99.10.2143/AST.29.1.504607Search in Google Scholar
Wang, P., Cockburn L. M., and Puterman M. L. 1998. “Analysis of Patent Data a Mixed-Poisson-Regression-Model Approach.” Journal of Business and Economic Statistics 16 (1): 27–41.10.1080/07350015.1998.10524732Search in Google Scholar
Wang, P., Puterman M. L., Cockburn I., and Le N. 1996. “Mixed Poisson Regression Models with Covariate Dependent Rates.” Biometrics 52 (2): 381–400.10.2307/2532881Search in Google Scholar
Wedel, M., DeSarbo W. S., Bult J. R., and Ramaswamy V. 1993. “A Latent Class Poisson Regression Model for Heterogeneous Count Data.” Journal of Applied Econometrics 8 (4): 397–411.10.1002/jae.3950080407Search in Google Scholar
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): 153–163.10.1016/j.insmatheco.2004.11.002Search in Google Scholar
Zamani, H., and Ismail N. 2010. “Negative Binomial-Lindley Distribution and its Application.” Journal of Mathematics and Statistics 6 (1): 4–9.10.3844/jmssp.2010.4.9Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
