Bayesian modeling of multivariate loss reserving data based on scale mixtures of multivariate normal distributions: estimation and case influence diagnostics

Abstract

One of the most important problems in general insurance is estimating the loss reserve distribution. In this article, we develop Bayesian multivariate loss reserving models for cases where losses and random effects are assumed to be distributed under the scale mixtures of multivariate normal (SMMN) distributions. This class of distributions, which contains heavy-tailed multivariate distributions such as student’s t, Pearson type VII, variance-gamma, slash and contaminated normal distributions, can be often used for robust inferences; when the assumptions of normality become questionable. The hierarchical structure of the SMMN representation has the advantage that under a Bayesian paradigm, the parameter estimation is simplified by sampling from multivariate normal distribution using Markov Chain Monte Carlo (MCMC) methods. A Bayesian case deletion influence diagnostics based on q-divergence measures is also presented. Further, simulated and real data sets are analyzed, where we show that the models under the Pearson type VII and the variance-gamma distributions outperform the usual normal models.