Robust Bayesian estimation of insurance premium in collective risk model

The collective risk model for the insurance claims is considered. The objective is to estimate a premium which is defined as a functional H specified up to an unknown parameter 'theta' (the expected number of claims). Four principles of calculating a premium are applied: net, variance principle, Esscher and exponential. The Bayesian methodology, which combines the prior knowledge about a parameter 'theta' with the knowledge in the form of a random sample, is adopted. Two loss functions (the square loss function and the asymmetric loss function LINEX) are considered. The obtained Bayes premium depends on a choice of a prior. Some uncertainty about a prior is assumed by introducing four classes of priors. The oscillation of the Bayes estimator is calculated. Considering one of the concepts of robust procedures the posterior regret 'Gamma' -minimax premiums are calculated as optimal robust premiums. A numerical example is presented.