Claim Reserving Using Distance-Based Generalized Linear Models
Generalized linear models (GLM) can be considered a stochastic version of the classical Chain-Ladder (CL) method of claim reserving in nonlife insurance. In particular, the deterministic CL model is reproduced when a GLM is fitted assuming over-dispersed
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Abstract Generalized linear models (GLM) can be considered a stochastic version of the classical Chain-Ladder (CL) method of claim reserving in nonlife insurance. In particular, the deterministic CL model is reproduced when a GLM is fitted assuming over-dispersed Poisson error distribution and logarithmic link. Our aim is to propose the use of distance-based generalized linear models (DB-GLM) in the claim reserving problem. DB-GLM can be considered a generalization of the classical GLM to the distance-based analysis, because DB-GLM contains as a particular instance ordinary GLM when the Euclidean, l 2 , metric is applied. Then, DB-GLM can be considered too a stochastic version of the CL claim reserving method. In DB-GLM, the only information required is a predictor distance matrix. DB-GLM can be fitted using the dbstats package for R. To estimate reserve distributions and standard errors, we propose a nonparametric bootstrap technique adequate to the distance-based regression models. We illustrate the method with a well-known actuarial dataset. Keywords Reserving · Chain-Ladder · Generalized linear models · Distance-based prediction · dbstats.
1 Introduction The objective of this work is to propose the DB-GLM [7] as an alternative methodology to solve the claim reserving problem. To complete the tool, we propose using the nonparametric technique of bootstrapping pairs [9] to estimate the predictive distribution of reserves. Bootstrapping pairs is an adequate bootstrap technique for DB-GLM as is proposed in [8]. E. Boj (B) · T. Costa Facultat d’Economia i Empresa, Universitat de Barcelona, Avinguda Diagonal 690, 08034 Barcelona, Spain e-mail: [email protected] T. Costa e-mail: [email protected] © Springer International Publishing Switzerland 2016 R. Cao et al. (eds.), Nonparametric Statistics, Springer Proceedings in Mathematics & Statistics 175, DOI 10.1007/978-3-319-41582-6_10
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To contextualize the claim reserving problem we list some of the pioneers deterministic methods in the literature (see [1, 20]): Grossing-up, Link ratio, CL, variants of CL, de Vylder least squares, and arithmetic and geometric separation methods of Taylor. Later some stochastic methods were proposed, both probabilistic and Bayesian. A list of some of those methods is: the Mack model, the BornhuetterFerguson model and the Munich Chain-Ladder method. The three methods have the common characteristic of generalizing (from a stochastic point of view) the CL deterministic method, because the estimation of the reserves coincides in all of them. The same occurs when using GLM (see [15]) as a stochastic method of claim reserving, with the assumptions of over-dispersed Poisson distribution and logarithmic link. In this case, the estimation of the reserves coincides with that of the CL deterministic method (see [14, 18, 21, 22] for a detailed proof). Additionally, the GLM has as particular cases other deterministic methods as are: the least squares of de Vylder and the arithmetic and geometric separation methods of Taylor. Th
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