A note on linear empirical Bayes estimation of finite population means

In this paper we consider linear empirical Bayes estimation of m finite population means with nj observations from the jth population and obtain generalizations of some of the theorems of Ghosh and Lahiri (J. Amer. Statist. Assoc. 82 (1987) 1153) to the case where njʹs are not necessarily bounded and may change with m. Our results strengthen, among other things, the estimators of Ghosh and Lahiri (1987) by showing asymptotical optimality of their estimators under more general conditions. They have also useful applications in credibility theory, which considers a similar problem with m populations that are not necessarily finite.