Deriving Unbiased Risk Estimators of Multinormal Matrix Mean Estimators Using Zonal Polynomials.

摘要

Unbiased risk estimators are derived for estimators in certain classes of equivariant estimators of multinormal matrix means, xi. In the case when the covariance structure is known these estimators are based on the sufficient statistic, X, a p x k matrix whose elements are normally distributed and for which E(X) = xi. In cases where the covariance is unknown, it is assumed that there is available independently observed data from which the covariance may be estimated. The method is a multivariate version of that introduced by James and Stein (1960) in establishing the worth of their estimator. The multivariate version uses known zonal polynomial expansions for the distributions of noncentral statistics to achieve the required generalization of the Pitman-Robbins (1949) representation of a noncentral chi-squared statistic.