On the Existence of Proper Bayes Minimax Estimators of the Mean of a Multivariate Normal Distribution.

摘要

Consider the problem of estimating the mean of a multivariate normal distribution with covariance matrix the identity and sum of squared errors loss. In an earlier paper the author showed that if the dimension p is 5or greater,then proper Bayes minimax estimators do exist. This result is reviewed briefly. The main purpose of the present paper is to show for p equal to 3or 4,that there do not exist spherically symmetric proper Bayes minimax estimators. The author has been unable,thus far,to disprove the existence of a nonspherical proper Bayes minimax estimator for p equal 3or 4. Of course,for p = 1,2,the usual estimator X bar is unique minimax but not proper Bayes. Bounds are derived for the possible bias of a minimax estimator. This result should be of some interest independent of its use in proving the main result of the paper. (Author)