Statistics Papers

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)=(δθ)tQ(δθ)/tr(QΣ̸), where Q is a known positive definite matrix. Estimators of the following form are considered:

δc(X,W)=(IcαQ−1W−1/(XtW−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)/( np−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.

Keywords

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

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Date Posted: 27 November 2017

This document has been peer reviewed.