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Bayesian CART models for insurance claims frequency

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  • Zhang, Yaojun
  • Ji, Lanpeng
  • Aivaliotis, Georgios
  • Taylor, Charles

Abstract

The accuracy and interpretability of a (non-life) insurance pricing model are essential qualities to ensure fair and transparent premiums for policy-holders, that reflect their risk. In recent years, classification and regression trees (CARTs) and their ensembles have gained popularity in the actuarial literature, since they offer good prediction performance and are relatively easy to interpret. In this paper, we introduce Bayesian CART models for insurance pricing, with a particular focus on claims frequency modelling. In addition to the common Poisson and negative binomial (NB) distributions used for claims frequency, we implement Bayesian CART for the zero-inflated Poisson (ZIP) distribution to address the difficulty arising from the imbalanced insurance claims data. To this end, we introduce a general MCMC algorithm using data augmentation methods for posterior tree exploration. We also introduce the deviance information criterion (DIC) for tree model selection. The proposed models are able to identify trees which can better classify the policy-holders into risk groups. Simulations and real insurance data will be used to illustrate the applicability of these models.

Suggested Citation

  • Zhang, Yaojun & Ji, Lanpeng & Aivaliotis, Georgios & Taylor, Charles, 2024. "Bayesian CART models for insurance claims frequency," Insurance: Mathematics and Economics, Elsevier, vol. 114(C), pages 108-131.
    • Handle: RePEc:eee:insuma:v:114:y:2024:i:c:p:108-131
    DOI: 10.1016/j.insmatheco.2023.11.005
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    References listed on IDEAS

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    1. Michel Denuit & Arthur Charpentier & Julien Trufin, 2021. "Autocalibration and Tweedie-dominance for Insurance Pricing with Machine Learning," Papers 2103.03635, arXiv.org, revised Jul 2021.
    2. Roel Henckaerts & Katrien Antonio & Maxime Clijsters & Roel Verbelen, 2018. "A data driven binning strategy for the construction of insurance tariff classes," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2018(8), pages 681-705, September.
    3. Jared S. Murray, 2021. "Log-Linear Bayesian Additive Regression Trees for Multinomial Logistic and Count Regression Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(534), pages 756-769, April.
    4. Christopher Blier-Wong & Hélène Cossette & Luc Lamontagne & Etienne Marceau, 2020. "Machine Learning in P&C Insurance: A Review for Pricing and Reserving," Risks, MDPI, vol. 9(1), pages 1-26, December.
    5. Denuit, Michel & Charpentier, Arthur & Trufin, Julien, 2021. "Autocalibration and Tweedie-dominance for insurance pricing with machine learning," LIDAM Discussion Papers ISBA 2021013, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Denuit, Michel & Charpentier, Arthur & Trufin, Julien, 2021. "Autocalibration and Tweedie-dominance for insurance pricing with machine learning," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 485-497.
    7. Antonio R. Linero & Debajyoti Sinha & Stuart R. Lipsitz, 2020. "Semiparametric mixed‐scale models using shared Bayesian forests," Biometrics, The International Biometric Society, vol. 76(1), pages 131-144, March.
    8. Denuit, Michel & Charpentier , Arthur & Trufin, Julien, 2021. "Autocalibration and Tweedie-dominance for insurance pricing with machine learning," LIDAM Reprints ISBA 2021049, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. David J. Spiegelhalter & Nicola G. Best & Bradley P. Carlin & Angelika Van Der Linde, 2002. "Bayesian measures of model complexity and fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 583-639, October.
    10. Hu, Changyue & Quan, Zhiyu & Chong, Wing Fung, 2022. "Imbalanced learning for insurance using modified loss functions in tree-based models," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 13-32.
    11. Antonio R. Linero, 2018. "Bayesian Regression Trees for High-Dimensional Prediction and Variable Selection," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 626-636, April.
    12. Yi Liu & Veronika Ročková & Yuexi Wang, 2021. "Variable selection with ABC Bayesian forests," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(3), pages 453-481, July.
    13. Roel Henckaerts & Marie-Pier Côté & Katrien Antonio & Roel Verbelen, 2021. "Boosting Insights in Insurance Tariff Plans with Tree-Based Machine Learning Methods," North American Actuarial Journal, Taylor & Francis Journals, vol. 25(2), pages 255-285, April.
    14. David J. Spiegelhalter & Nicola G. Best & Bradley P. Carlin & Angelika Linde, 2014. "The deviance information criterion: 12 years on," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(3), pages 485-493, June.
    15. Meng, Shengwang & Gao, Yaqian & Huang, Yifan, 2022. "Actuarial intelligence in auto insurance: Claim frequency modeling with driving behavior features and improved boosted trees," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 115-127.
    16. Antonio R. Linero & Yun Yang, 2018. "Bayesian regression tree ensembles that adapt to smoothness and sparsity," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(5), pages 1087-1110, November.
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    More about this item

    Keywords

    Bayesian CART; Claims frequency; DIC; Insurance pricing; MCMC; Negative binomial distribution; Zero-inflated Poisson distribution;
    All these keywords.

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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