Estimators for Gaussian Models Having a Block-wise Structure

Many multivariate Gaussian models can conveniently be split into independent, block-wise problems. Common settings where this situation arises are balanced ANOVA models, balanced longitudinal models, and certain block-wise shrinkage estimators in nonparametric regression estimation involving orthogonal bases such as Fourier or wavelet bases. It is well known that the standard, least squares estimate in multidimensional Gaussian models can often be improved through the use of minimax shrinkage estimators or related Bayes estimators. In the following we show that the traditional estimators constructed via independent shrinkage can be improved in terms of their squared-error risk, and we provide improved minimax estimators. An alternate class of block-wise shrinkage estimators is also considered, and fairly precise conditions are given that characterize when these estimators are admissible or quasi-admissible. These results can also be applied to the classical Stein-Lindley estimator that shrinks toward an overall mean. It is shown how this estimator can be improved by introducing additional shrinkage.