On asymptotic optimality in empirical Bayes credibility

In this paper we consider linear empirical Bayes estimation of the pure premiums in a Bühlmann–Straub model, with m risk contracts and ni observations from the ith contract, and provide sufficient conditions, analogous to the conditions of Norberg [Scandinavian Actuarial Journal (1980) 172–194], for a stricter notion of asymptotic optimality. The sufficient conditions are used in the paper to show asymptotic optimality of explicit non-parametric examples. Our results, among other things, strengthen the estimators proposed by Bühlmann and Straub (1970) and their generalization by Sundt (1983). They also strengthen the finite population mean estimators of Ghosh and Lahiri (1987) by showing the asymptotic optimality of their estimators under a weaker condition than the boundedness of the moments of fourth order.