Skew mixture models for loss distributions: a Bayesian approach

The derivation of loss distribution from insurance data is a very interesting research topic but at the same time not an easy task. To find an analytic solution to the loss distribution may be mislading although this approach is frequently adopted in the actuarial literature. Moreover, it is well recognized that the loss distribution is strongly skewed with heavy tails and present small, medium and large size claims which hardly can be fitted by a single analytic and parametric distribution. Here we propose a finite mixture of Skew Normal distributions that provides a better characterization of insurance data. We adopt a Bayesian approach to estimate the model, providing the likelihood and the priors for the all unknow parameters; we implement an adaptive Markov Chain Monte Carlo algorithm to approximate the posterior distribution. We apply our approach to a well known Danish fire loss data and relevant risk measures, as Value-at-Risk and Expected Shortfall probability, are evaluated as well.