Bayes estimates as expanders in one and two dimensions

Consider the problem of estimating a p-variate normal mean using squared error loss. We demonstrate that contrary to what one might expect, (a) sup E{parallel tod((X) under tilde)parallel to/parallel to} > 1 for p less than or equal to 2, where the supremum is over all priors G symmetric about (0) under tilde, E((.)) denotes marginal expectation corresponding to G and d((X) under tilde) denotes Bayes rule with respect to G. (b) The above supremum is equal to 1 for all p greater than or equal to 3. Thus Bayes estimates corresponding to symmetric priors can be expanders only in one and two dimensions. For p = 1,2, the numerical values of the supremum are obtained. We also prove that sup E{parallel tod((X) under tilde)parallel to/parallel to (X) under tilde parallel to} = 1 for every p > 1 if one restricts to spherically symmetric priors. In the process, some closed form formulas of independent interest are also obtained. (C) 2002 Elsevier Science B.V. All rights reserved.