Minimax estimation of a multivariate normal mean under polynomial loss

Let X be an observation from a p-variate (p >= 3) normal random vector with unknown mean vector [theta] and known covariance matrix . The problem of improving upon the usual estimator of [theta], [delta]0(X) = X, is considered. An approach is developed which can lead to improved estimators, [delta], for loss functions which are polynomials in the coordinates of ([delta] - [theta]). As an example of this approach, the loss L([delta], [theta]) = [delta] - [theta]4 is considered, and estimators are developed which are significantly better than [delta]0. When is the identity matrix, these estimators are of the form [delta](X) = (1 - (b/(d + X2)))X.