Employing vague prior information in the construction of confidence sets

In the problem of estimating the mean, θ, of a multivariate normal distribution, an experimenter will often be able to give some vague prior specifications about θ. This information is used to construct confidence sets centered at improved estimators of θ. These sets are shown to have uniformly (in θ) higher coverage probability than the usual confidence set (a sphere centered at the observations), with no increase in volume. Further, through the use of a modified empirical Bayes argument, a variable radius confidence set is constructed which provides a uniform reduction of volume. Strong numerical evidence is presented which shows that the empirical Bayes set also dominates the usual confidence set in coverage probability. All these improved sets provide substantial gains if the prior information is correct. Also considered are extensions to the unknown variance case, and a discussion of applications to the one-way analysis of variance. In particular, a procedure is presented which uniformly improves upon Scheffé's method of estimation of contrasts.