Robust Bayesian estimation and prediction in gamma-gamma model of claim reserves

The problem of robust Bayesian estimation of chain ladder (development) factors and Bayesian prediction of claim reserves is considered. Two different classes of priors (the classes, where parameters of a prior are not specified exactly and the class, where the prior cumulative distribution function is distorted) are presented. The oscillation (as a measure of robustness) of Bayes estimators and predictors of reserves, when priors are in the considered class, is calculated and the posterior regret Γ-minimax (PRGM) rules as optimal procedures are obtained. The numerical example compares different methods of the estimation of development factors and the calculation of claim reserves. The chain ladder estimators and predictors, exact Bayes estimators and predictors, PRGM estimators and predictors for aforementioned classes of priors and empirical credibility estimators and predictors are considered. It is shown that the variability of the expected value parameter of a prior has a greater impact on the oscillation of the Bayes estimators of the development factors and Bayes predictors of reserves than the variability of the shape parameter. The Bayes predictors are more robust with respect to the distortion of a prior c.d.f. than the fluctuation of the expected value parameter of a Gamma prior. A distortion of the shape of the prior c.d.f. has also a smaller impact on the value of PRGM estimators and predictors than the variability of the parameters of Gamma distribution. The difference between the Bayesian and chain ladder estimators (and predictors) depends mainly on the difference between the expected value parameter of the prior and the chain ladder estimator of development factor.