Bayesian analysis in an aggregate loss model: validation of the structure functions

The aggregate loss, S, is the sum of the individual claim sizes, ie, S=∑i=1KXi for K>0, and S=0 for K=0. The distributions of K and Xi are termed the primary distribution and the secondary distribution, respectively. This paper considers the empirical evaluation of a collective risk model with the geometric as the primary distribution and the exponential as the secondary distribution. We develop a Bayesian analysis for three risk variables (number of claims, severity and aggregate loss), and apply it to sixteen real portfolios of policyholders, to illustrate the validity of the model. The net premium is obtained for each variable and the results are compared with those derived from a frequentist approach given by maximum likelihood estimation. From a Bayesian standpoint, the interest of this study lies in its specification of the prior distributions (structure functions) and in how the hyperparameters are elicited. In order to validate the specification of the prior information, we consider two different distributions for structure functions (the beta and uniform distributions) and, in each case, study the determination of the hyperparameters. We analyze the moment and maximum likelihood methods and propose a new possibility based on prior information specified by an expert in terms of quantiles, suggesting three different scenarios. These comparisons show that in all cases the Bayes premium is practically equal to the net premium with the parameter estimated by maximum likelihood. This finding indicates that in nonlife insurance problems the importance of the data, modeled by the likelihood, outweighs the prior information.