Issues in Claims Reserving and Credibility: a Semiparametric Approach with Mixed Models

Verrall (1996) and England & Verrall (2001) first considered the use of smoothing methods in the context of claims reserving. They applied two smoothing procedures in a likelihood-based way, namely the locally weighted regression smoother (‘loess’) and the cubic smoothing spline smoother. Using the statistical methodology of semiparametric regression and its connection with mixed models (see e.g. Ruppert et al., 2003), this paper revisits smoothing models for loss reserving and credibility. Apart from the flexibility inherent to all semiparametric methods, advantages of the semiparametric approach developed here are threefold. Firstly, a Bayesian implementation of these smoothing models is relatively straightforward and allows simulation from the full predictive distribution of quantities of interest. Since the main interest of actuaries lies in prediction, this is a major advantage. Secondly, because the constructed models have an interpretation as (generalized) linear mixed models ((G)LMMs), standard statistical theory and software for (G)LMMs can be used. Thirdly, more complicated data sets, dealing for example with quarterly development in a reserving context, heavy-tails, semicontinuous data, or extensive longitudinal data, can be modelled within this framework. Throughout this article, data examples illustrate these different aspects. Several comments are included regarding model specification, estimation and selection.