ScholarlyCommons

Title

Minimax Estimation of a Normal Mean Vector for Arbitrary Quadratic Loss and Unknown Covariance Matrix

Author(s)

J. Berger
M. E. Bock
Lawrence D. Brown, University of Pennsylvania
George Casella
L. Gleser

Document Type

Journal Article

Date of this Version

1977

Publication Source

The Annals of Statistics

Volume

5

Issue

4

Start Page

763

Last Page

771

DOI

10.1214/aos/1176343898

Abstract

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:

δ^c(X,W)=(I−cαQ⁻¹W⁻¹/(X^tW⁻¹X))X,

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,δ^c is shown to be minimax and better than the usual estimator δ⁰(X)=X.

Keywords

minimax, normal, mean, quadratic loss, unknown covariance matrix, Wishart, risk function

Recommended Citation

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