Two-Stage Sequential Estimation of a Multivariate Normal Mean under Quadratic Loss

Abstract

In estimating a multivariate normal mean under quadratic loss, this paper looks into the existence of two-stage sequential estimators that are better both in risk (mean square error) and sample size than the usual estimator of a given fixed sample size. In other words, given any sample size $n$, we are looking for two-stage sequential estimators truncated at $n$, with a positive probability of stopping earlier and risk lower than that of the sample mean based on $n$ observations. Sequential versions of James-Stein estimators are used to produce two-stage sequential estimators better in risk and sample size than the usual estimator--the sample mean. A lower bound on the largest possible probability of stopping earlier without losing in the risk is also obtained.