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.
"Deriving Unbiased Risk Estimators of Multinormal Mean and Regression Coefficient Estimators Using Zonal Polynomials." Ann. Statist. 6 (4) 769 - 782, July, 1978. https://doi.org/10.1214/aos/1176344251