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 θ and known covariance matrix {A figure is presented}. The problem of improving upon the usual estimator of θ, δ0(X) = X, is considered. An approach is developed which can lead to improved estimators, δ, for loss functions which are polynomials in the coordinates of (δ - θ). As an example of this approach, the loss L(δ, θ) = |δ - θ|4 is considered, and estimators are developed which are significantly better than δ0. When {A figure is presented} is the identity matrix, these estimators are of the form δ(X) = (1 - ( b (d + |X|2)))X. © 1978.

Duke Scholars

Author James O. Berger Statistical Science