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

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.