Improved estimation of covariance matrices in balanced hierarchical multivariate variance components models

The problem of simultaneous estimation of covariance matrices in balanced hierarchical multivariate variance components models is considered. A new class of estimators is proposed which dominates the usual sensible estimators with respect to total variability (sum of squared error losses). These estimators shrink towards a multiple of an identity matrix, the multiple being the geometric mean of the characteristic roots of the component Wishart matrices. Numerical illustrations are considered to exhibit the improvement in risk under a simple model.