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Bivariate Mixed Poisson and Normal Generalised Linear Models with Sarmanov Dependence—An Application to Model Claim Frequency and Optimal Transformed Average Severity

Author

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  • Ramon Alemany

    (Department of Econometrics, Statistics and Applied Economics, Riskcenter-IREA University of Barcelona, Av. Diagonal, 690, 08034 Barcelona, Spain
    These authors contributed equally to this work.)

  • Catalina Bolancé

    (Department of Econometrics, Statistics and Applied Economics, Riskcenter-IREA University of Barcelona, Av. Diagonal, 690, 08034 Barcelona, Spain
    These authors contributed equally to this work.)

  • Roberto Rodrigo

    (Department of Econometrics, Statistics and Applied Economics, Riskcenter-IREA University of Barcelona, Av. Diagonal, 690, 08034 Barcelona, Spain
    These authors contributed equally to this work.)

  • Raluca Vernic

    (Faculty of Mathematics and Computer Science, Ovidius University of Constanta, 124 Mamaia, Constanta, Romania
    Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie 13, 050711 Bucharest, Romania
    These authors contributed equally to this work.)

Abstract

The aim of this paper is to introduce dependence between the claim frequency and the average severity of a policyholder or of an insurance portfolio using a bivariate Sarmanov distribution, that allows to join variables of different types and with different distributions, thus being a good candidate for modeling the dependence between the two previously mentioned random variables. To model the claim frequency, a generalized linear model based on a mixed Poisson distribution -like for example, the Negative Binomial (NB), usually works. However, finding a distribution for the claim severity is not that easy. In practice, the Lognormal distribution fits well in many cases. Since the natural logarithm of a Lognormal variable is Normal distributed, this relation is generalised using the Box-Cox transformation to model the average claim severity. Therefore, we propose a bivariate Sarmanov model having as marginals a Negative Binomial and a Normal Generalized Linear Models (GLMs), also depending on the parameters of the Box-Cox transformation. We apply this model to the analysis of the frequency-severity bivariate distribution associated to a pay-as-you-drive motor insurance portfolio with explanatory telematic variables.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2020:i:1:p:73-:d:472799
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    References listed on IDEAS

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