## The Annals of Statistics

### Deriving Unbiased Risk Estimators of Multinormal Mean and Regression Coefficient Estimators Using Zonal Polynomials

Jim Zidek

#### Abstract

Unbiased risk estimators are derived for estimators in certain classes of equivariant estimators of multinormal matrix means, $\xi,$ and regression coefficients $\beta.$ In all cases the covariance matrix is unknown. The underlying method, a multivariate version of that of James and Stein (1960), uses zonal polynomial expansions for the distributions of noncentral statistics. This gives, in one case, the required generalization of the Pitman-Robbins representation of noncentral chi-square statistics including the appropriate multivariate Poisson law. In the other case, a multivariate negative binomial law emerges. The result for regression coefficients suggests a new minimax estimator and, essentially, an extension of Baranchik's result.

#### Article information

Source
Ann. Statist., Volume 6, Number 4 (1978), 769-782.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176344251

Digital Object Identifier
doi:10.1214/aos/1176344251

Mathematical Reviews number (MathSciNet)
MR478428

Zentralblatt MATH identifier
0379.62008

JSTOR