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A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking

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  • Bermúdez, Lluís
  • Karlis, Dimitris

Abstract

Bivariate Poisson regression models for ratemaking in car insurance have been previously used. They included zero-inflated models to account for the excess of zeros and the overdispersion in the data set. These models are now revisited in order to consider alternatives. A 2-finite mixture of bivariate Poisson regression models is used to demonstrate that the overdispersion in the data requires more structure if it is to be taken into account, and that a simple zero-inflated bivariate Poisson model does not suffice. At the same time, it is shown that a finite mixture of bivariate Poisson regression models embraces zero-inflated bivariate Poisson regression models as a special case. Finally, an EM algorithm is provided in order to ensure the models’ ease-of-fit. These models are applied to an automobile insurance claims data set and it is shown that the modeling of the data set can be improved considerably.

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  • Bermúdez, Lluís & Karlis, Dimitris, 2012. "A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 3988-3999.
    • Handle: RePEc:eee:csdana:v:56:y:2012:i:12:p:3988-3999
    DOI: 10.1016/j.csda.2012.05.016
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    Cited by:

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    3. Spark C. Tseung & Ian Weng Chan & Tsz Chai Fung & Andrei L. Badescu & X. Sheldon Lin, 2022. "A Posteriori Risk Classification and Ratemaking with Random Effects in the Mixture-of-Experts Model," Papers 2209.15212, arXiv.org.
    4. Delong, Łukasz & Lindholm, Mathias & Wüthrich, Mario V., 2021. "Gamma Mixture Density Networks and their application to modelling insurance claim amounts," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 240-261.
    5. Zezhun Chen & Angelos Dassios & George Tzougas, 2023. "Multivariate mixed Poisson Generalized Inverse Gaussian INAR(1) regression," Computational Statistics, Springer, vol. 38(2), pages 955-977, June.
    6. Tatjana Miljkovic & Daniel Fernández, 2018. "On Two Mixture-Based Clustering Approaches Used in Modeling an Insurance Portfolio," Risks, MDPI, vol. 6(2), pages 1-18, May.
    7. Zezhun Chen & Angelos Dassios & George Tzougas, 2022. "EM Estimation for the Bivariate Mixed Exponential Regression Model," Risks, MDPI, vol. 10(5), pages 1-13, May.
    8. Tzougas, George & Pignatelli di Cerchiara, Alice, 2021. "The multivariate mixed Negative Binomial regression model with an application to insurance a posteriori ratemaking," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 602-625.
    9. Počuča, Nikola & Jevtić, Petar & McNicholas, Paul D. & Miljkovic, Tatjana, 2020. "Modeling frequency and severity of claims with the zero-inflated generalized cluster-weighted models," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 79-93.
    10. Yang Lu, 2018. "Dynamic Frailty Count Process in Insurance: A Unified Framework for Estimation, Pricing, and Forecasting," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 85(4), pages 1083-1102, December.
    11. Lluís Bermúdez & Dimitris Karlis & Isabel Morillo, 2020. "Modelling Unobserved Heterogeneity in Claim Counts Using Finite Mixture Models," Risks, MDPI, vol. 8(1), pages 1-13, January.
    12. Pechon, Florian & Denuit, Michel & Trufin, Julien, 2018. "Multivariate Modelling of Multiple Guarantees in Motor Insurance of a Household," LIDAM Discussion Papers ISBA 2018019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    13. George Tzougas & Despoina Makariou, 2022. "The multivariate Poisson‐Generalized Inverse Gaussian claim count regression model with varying dispersion and shape parameters," Risk Management and Insurance Review, American Risk and Insurance Association, vol. 25(4), pages 401-417, December.
    14. Bermúdez, Lluís & Guillén, Montserrat & Karlis, Dimitris, 2018. "Allowing for time and cross dependence assumptions between claim counts in ratemaking models," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 161-169.
    15. Su Pei-Fang & Mau Yu-Lin & Li Chung-I & Guo Yan & Liu Qi & Shyr Yu & Boice John D., 2017. "Bivariate Poisson models with varying offsets: an application to the paired mitochondrial DNA dataset," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 16(1), pages 47-58, March.
    16. Tzougas, George & Makariou, Despoina, 2022. "The multivariate Poisson-Generalized Inverse Gaussian claim count regression model with varying dispersion and shape parameters," LSE Research Online Documents on Economics 117197, London School of Economics and Political Science, LSE Library.
    17. Lluís Bermúdez & Dimitris Karlis, 2022. "Copula-based bivariate finite mixture regression models with an application for insurance claim count data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(4), pages 1082-1099, December.
    18. Gning, Lucien & Diagne, M.L. & Tchuenche, J.M., 2023. "Hierarchical generalized linear models, correlation and a posteriori ratemaking," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 614(C).

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