Date of this Version
The Annals of Statistics
Let X be an observation from a p-variate normal distribution (p ≧ 3) with mean vector θ and unknown positive definite covariance matrix Σ̸. It is desired to estimate θ under the quadratic loss L(δ,θ,Σ̸)=(δ−θ)tQ(δ−θ)/tr(QΣ̸), where Q is a known positive definite matrix. Estimators of the following form are considered:
where W is a p × p random matrix with a Wishart (Σ̸,n) distribution (independent of X), α is the minimum characteristic root of (QW)/( n−p−1) and c is a positive constant. For appropriate values of c,δc is shown to be minimax and better than the usual estimator δ0(X)=X.
minimax, normal, mean, quadratic loss, unknown covariance matrix, Wishart, risk function
Berger, J., Bock, M. E., Brown, L. D., Casella, G., & Gleser, L. (1977). Minimax Estimation of a Normal Mean Vector for Arbitrary Quadratic Loss and Unknown Covariance Matrix. The Annals of Statistics, 5 (4), 763-771. http://dx.doi.org/10.1214/aos/1176343898
Date Posted: 27 November 2017
This document has been peer reviewed.