Empirical and Hierarchical Bayes Competitors of Preliminary Test Estimators in Two Sample Problems

摘要

Suppose in a laboratory, say Laboratory I, a certain instrument is designed to measure several characteristics, and a number of vector-valued measurements is recorded. The objective is to estimate the unknown population mean. It is known, however, that a similar instrument is used in another laboratory, say Laboratory II for the same purpose, and a number of observations is recorded from the second instrument. It is also suspected that the two population means are equal, in which case, observations recorded in Laboratory II can possibly be used effectively together with those in Laboratory I for estimating the population mean of the first instrument. Thus, the question that naturally arises is whether one should use the sample mean from Laboratory I or the pooled mean from the two laboratories. In problems of this type what is normally sought is a compromise estimator which leans more towards the pooled sample mean when the null hypothesis of the equality of the two population means is accepted, and towards the sample mean from Laboratory I when such a hypothesis is rejected. A very popular way to achieve this compromise is to use a preliminary test estimator (PTE) which uses the pooled mean when the null hypothesis is accepted at a desired level of significance, and uses the sample mean from Laboratory I when opposite is the case. This paper proposes instead an empirical Bayes estimator which achieves the intended compromise.