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Gamma Mixture Density Networks and their application to modelling insurance claim amounts

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  • Delong, Łukasz
  • Lindholm, Mathias
  • Wüthrich, Mario V.

Abstract

We discuss how mixtures of Gamma distributions with mixing probabilities, shape and rate parameters depending on features can be fitted with neural networks. We develop two versions of the EM algorithm for fitting so-called Gamma Mixture Density Networks, which we call the EM network boosting algorithm and the EM forward network algorithm, and we test their implementation together with the choices of hyperparameters. A simulation study shows that our algorithms perform very well on synthetic data sets. We further illustrate the application of the Gamma Mixture Density Network on a real data set of motor insurance claim amounts and conclude that Gamma Mixture Density Networks can improve the fit of the regression model and the predictions of the claim severities used for rate-making compared to classical actuarial techniques.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:insuma:v:101:y:2021:i:pb:p:240-261
    DOI: 10.1016/j.insmatheco.2021.08.003
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    1. Parodi, Pietro, 2020. "A Generalised Property Exposure Rating Framework That Incorporates Scale-Independent Losses And Maximum Possible Loss Uncertainty," ASTIN Bulletin, Cambridge University Press, vol. 50(2), pages 513-553, May.
    2. Tzougas, George & Karlis, Dimitris, 2020. "An Em Algorithm For Fitting A New Class Of Mixed Exponential Regression Models With Varying Dispersion," ASTIN Bulletin, Cambridge University Press, vol. 50(2), pages 555-583, May.
    3. Simon Lee & X. Lin, 2010. "Modeling and Evaluating Insurance Losses Via Mixtures of Erlang Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 14(1), pages 107-130.
    4. Miljkovic, Tatjana & Grün, Bettina, 2016. "Modeling loss data using mixtures of distributions," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 387-396.
    5. 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.
    6. Blostein, Martin & Miljkovic, Tatjana, 2019. "On modeling left-truncated loss data using mixtures of distributions," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 35-46.
    7. Michel Wedel & Wayne DeSarbo, 1995. "A mixture likelihood approach for generalized linear models," Journal of Classification, Springer;The Classification Society, vol. 12(1), pages 21-55, March.
    8. 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.
    9. Tzougas, George & Karlis, Dimitris, 2020. "An EM algorithm for fitting a new class of mixed exponential regression models with varying dispersion," LSE Research Online Documents on Economics 104027, London School of Economics and Political Science, LSE Library.
    10. Derek S. Young & Xi Chen & Dilrukshi C. Hewage & Ricardo Nilo-Poyanco, 2019. "Finite mixture-of-gamma distributions: estimation, inference, and model-based clustering," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 13(4), pages 1053-1082, December.
    11. 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.
    12. Christopher N Davis & T Deirdre Hollingsworth & Quentin Caudron & Michael A Irvine, 2020. "The use of mixture density networks in the emulation of complex epidemiological individual-based models," PLOS Computational Biology, Public Library of Science, vol. 16(3), pages 1-16, March.
    13. Verbelen, Roel & Gong, Lan & Antonio, Katrien & Badescu, Andrei & Lin, Sheldon, 2015. "Fitting Mixtures Of Erlangs To Censored And Truncated Data Using The Em Algorithm," ASTIN Bulletin, Cambridge University Press, vol. 45(3), pages 729-758, September.
    14. Fung, Tsz Chai & Badescu, Andrei L. & Lin, X. Sheldon, 2019. "A class of mixture of experts models for general insurance: Theoretical developments," Insurance: Mathematics and Economics, Elsevier, vol. 89(C), pages 111-127.
    15. Yin, Cuihong & Lin, X. Sheldon, 2016. "EFFICIENT ESTIMATION OF ERLANG MIXTURES USING iSCAD PENALTY WITH INSURANCE APPLICATION," ASTIN Bulletin, Cambridge University Press, vol. 46(3), pages 779-799, September.
    16. Chai Fung, Tsz & Badescu, Andrei L. & Sheldon Lin, X., 2019. "A Class Of Mixture Of Experts Models For General Insurance: Application To Correlated Claim Frequencies," ASTIN Bulletin, Cambridge University Press, vol. 49(3), pages 647-688, September.
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    More about this item

    Keywords

    Expectation-maximization algorithm; Neural networks; Mixtures of distributions; Regression models; Rate-making;
    All these keywords.

    JEL classification:

    • C45 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Neural Networks and Related Topics
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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