Equivariant two-stage estimation of a noral mean

Abstract

We consider estimation of a normal mean based on a sample of independent identically distributed observations from a normal population with unknown variance. A two stage procedure involves first taking an initial sample, on the basis of that sample deciding on the size of a second sample, and then estimating the mean based on the total sample. The loss is a linear combination of mean squared error and number of observations taken. We restrict attention to procedures which are shift invariant. We show that a complete class of procedures consists of those which choose the second sample size monotonically in the initial sample variance, obtain a class of natural Generalized Bayes procedures and an asymptotically minimax procedure among procedures with given initial sample size, and show that this latter procedure is admissible among the full class of equivariant procedures.