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

Ramon Alemany, Catalina Bolancé, Roberto Rodrigo and Raluca Vernic
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Ramon Alemany: Department of Econometrics, Statistics and Applied Economics, Riskcenter-IREA University of Barcelona, Av. Diagonal, 690, 08034 Barcelona, Spain
Catalina Bolancé: Department of Econometrics, Statistics and Applied Economics, Riskcenter-IREA University of Barcelona, Av. Diagonal, 690, 08034 Barcelona, Spain
Roberto Rodrigo: Department of Econometrics, Statistics and Applied Economics, Riskcenter-IREA University of Barcelona, Av. Diagonal, 690, 08034 Barcelona, Spain
Raluca Vernic: Faculty of Mathematics and Computer Science, Ovidius University of Constanta, 124 Mamaia, Constanta, Romania

Mathematics, 2020, vol. 9, issue 1, 1-18

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.

Keywords: Box-Cox transformation; dependence; bivariate Sarmanov distribution; motor insurance; telematic data (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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