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Sarmanov family of multivariate distributions for bivariate dynamic claim counts model

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  • Abdallah, Anas
  • Boucher, Jean-Philippe
  • Cossette, Hélène

Abstract

To predict future claims, it is well-known that the most recent claims are more predictive than older ones. However, classic panel data models for claim counts, such as the multivariate negative binomial distribution, do not put any time weight on past claims. More complex models can be used to consider this property, but often need numerical procedures to estimate parameters. When we want to add a dependence between different claim count types, the task would be even more difficult to handle. In this paper, we propose a bivariate dynamic model for claim counts, where past claims experience of a given claim type is used to better predict the other type of claims. This new bivariate dynamic distribution for claim counts is based on random effects that come from the Sarmanov family of multivariate distributions. To obtain a proper dynamic distribution based on this kind of bivariate priors, an approximation of the posterior distribution of the random effects is proposed. The resulting model can be seen as an extension of the dynamic heterogeneity model described in Bolancé et al. (2007). We apply this model to two samples of data from a major Canadian insurance company, where we show that the proposed model is one of the best models to adjust the data. We also show that the proposed model allows more flexibility in computing predictive premiums because closed-form expressions can be easily derived for the predictive distribution, the moments and the predictive moments.

Suggested Citation

  • Abdallah, Anas & Boucher, Jean-Philippe & Cossette, Hélène, 2016. "Sarmanov family of multivariate distributions for bivariate dynamic claim counts model," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 120-133.
  • Handle: RePEc:eee:insuma:v:68:y:2016:i:c:p:120-133
    DOI: 10.1016/j.insmatheco.2016.01.003
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Catalina Bolancé & Raluca Vernic, 2020. "Frequency and Severity Dependence in the Collective Risk Model: An Approach Based on Sarmanov Distribution," Mathematics, MDPI, vol. 8(9), pages 1-17, August.
    2. Vernic, Raluca & Bolancé, Catalina & Alemany, Ramon, 2022. "Sarmanov distribution for modeling dependence between the frequency and the average severity of insurance claims," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 111-125.
    3. 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.
    4. 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.
    5. Bolancé, Catalina & Vernic, Raluca, 2019. "Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 89-103.
    6. Chen, Zezhun & Dassios, Angelos & Tzougas, George, 2022. "Multivariate mixed Poisson Generalized Inverse Gaussian INAR(1) regression," LSE Research Online Documents on Economics 115369, London School of Economics and Political Science, LSE Library.
    7. Ramon Alemany & Catalina Bolancé & Roberto Rodrigo & Raluca Vernic, 2020. "Bivariate Mixed Poisson and Normal Generalised Linear Models with Sarmanov Dependence—An Application to Model Claim Frequency and Optimal Transformed Average Severity," Mathematics, MDPI, vol. 9(1), pages 1-18, December.
    8. Denuit, Michel & Lu, Yang, 2020. "Wishart-Gamma mixtures for multiperil experience ratemaking, frequency-severity experience rating and micro-loss reserving," LIDAM Discussion Papers ISBA 2020016, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. 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.
    10. Michel Denuit & Yang Lu, 2021. "Wishart‐gamma random effects models with applications to nonlife insurance," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 88(2), pages 443-481, June.
    11. Youn Ahn, Jae & Jeong, Himchan & Lu, Yang, 2021. "On the ordering of credibility factors," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 626-638.
    12. Catalina Bolancé & Raluca Vernic, 2017. "“Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution”," IREA Working Papers 201718, University of Barcelona, Research Institute of Applied Economics, revised Oct 2017.
    13. Chen, Zezhun & Dassios, Angelos & Tzougas, George, 2022. "EM estimation for the bivariate mixed exponential regression model," LSE Research Online Documents on Economics 115132, London School of Economics and Political Science, LSE Library.
    14. Anas Abdallah & Lan Wang, 2023. "Rank-Based Multivariate Sarmanov for Modeling Dependence between Loss Reserves," Risks, MDPI, vol. 11(11), pages 1-37, October.
    15. 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.
    16. Tzougas, George & di Cerchiara, Alice Pignatelli, 2021. "Bivariate mixed Poisson regression models with varying dispersion," LSE Research Online Documents on Economics 114327, London School of Economics and Political Science, LSE Library.
    17. 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.
    18. Chen, Zezhun Chen & Dassios, Angelos & Tzougas, George, 2024. "EM estimation for bivariate mixed poisson INAR(1) claim count regression models with correlated random effects," LSE Research Online Documents on Economics 118826, London School of Economics and Political Science, LSE Library.
    19. Bingzhen Geng & Yang Liu & Yimiao Zhao, 2024. "Value-at-Risk- and Expectile-based Systemic Risk Measures and Second-order Asymptotics: With Applications to Diversification," Papers 2404.18029, arXiv.org.
    20. 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.

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