Dirichlet process mixture models for insurance loss data

Abstract

In the recent insurance literature, a variety of finite-dimensional parametric models have been proposed for analyzing the hump-shaped, heavy-tailed, and highly skewed loss data often encountered in applications. These parametric models are relatively simple, but they lack flexibility in the sense that an actuary analyzing a new data-set cannot be sure that any one of these parametric models will be appropriate. As a consequence, the actuary must make a non-trivial choice among a collection of candidate models, putting him/herself at risk for various model misspecification biases. In this paper, we argue that, at least in cases where prediction of future insurance losses is the ultimate goal, there is reason to consider a single but more flexible nonparametric model. We focus here on Dirichlet process mixture models, and we reanalyze several of the standard insurance data-sets to support our claim that model misspecification biases can be avoided by taking a nonparametric approach, with little to no cost, compared to existing parametric approaches.

Keywords:

• Danish fire losses data
• general insurance
• model misspecification
• nonparametric Bayes
• Norwegian fire losses data
• property and casualty insurance
• US allocated loss adjustment expenses data

Notes

No potential conflict of interest was reported by the authors.

1 Note that this is different from the classical notion of ‘nonparametric’ as in Lehmann (2006), where the goal is inference on a finite-dimensional quantity, such as a distribution quantile, under minimal conditions on the distribution itself. Here we are interested in the distribution itself, an infinite-dimensional object.