Bootstrapping Generalized Linear Models for Development Triangles Using Deviance Residuals

This paper presents a practical study of how to bootstrap a development triangle using a generalized linear model (GLM) and deviance residuals. We also point out some limitations inherent in bootstrapping approaches. (Interested readers can contact the author and request a copy of an MS Excel application to further explore the concepts discussed in this paper.) First we demonstrate how Pearson residual bootstrapping can fail when applied to GLMs because of their linear rescaling properties. Next we describe an algorithm for rescaling deviance residuals based on the identity variance function. We continue with an example where Pearson residual bootstrapping fails, while deviance residuals bootstrapping works. We then present bootstrap simulation results for two GLMs: one where both approaches work and the original example where only deviance residuals can be applied. Subsequently we prove that deviance residuals based on the identity variance function are bounded below for any given data point with the lower bound depending on the fitted value for the data point. We then give an example of a GLM where deviance residual bootstrapping fails because of this property. The paper concludes with a discussion “distribution-free” versus parametric resampling.

Keywords: Bootstrapping and Resampling Methods, Generalized Linear Modeling, Reserve Variability, Reserving Methods, Nonparametric Methods.