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{1176344251,
author = {Zidek, Jim},
title = {Deriving Unbiased Risk Estimators of Multinormal Mean and Regression Coefficient Estimators Using Zonal Polynomials},
journal = {Ann. Statist.},
volume = {6},
number = {1},
year = {1978},
pages = { 769-782},
language = {en},
url = {http://dml.mathdoc.fr/item/1176344251}
}
Zidek, Jim. Deriving Unbiased Risk Estimators of Multinormal Mean and Regression Coefficient Estimators Using Zonal Polynomials. Ann. Statist., Tome 6 (1978) no. 1, pp. 769-782. http://gdmltest.u-ga.fr/item/1176344251/