Empirical Bayes estimation of the mean in a multivariate normal distribution

This Note, reprinted from Communications in Statistics, Theory and Methods, v. 15, no. 7, 1986, considers the problem of estimating the mean vector of a multivariate normal distribution under a variety of assumed structures among the parameters of the sampling and prior distributions. The authors use a pragmatic approach. They adopt prior distributional families, assess hyperparameters, and adopt patterned mean and covariance structures when it is relatively simple to do so; alternatively, they use the sample data to estimate hyperparameters of prior distributions when assessment is a formidable task (e.g., when assessing parameters of multidimensional problems). James-Stein-like estimators result. In some cases, the authors have been able to show that the estimators proposed uniformly dominate the maximum likelihood estimators when measured with respect to quadratic loss functions.