SURE estimates under dependence and heteroscedasticity

The multivariate Bayesian hierarchical model with independent means has been studied extensively and is widely used in practice. In contrast, the case of dependent means has received scant attention, even though multivariate observations are often correlated. In this paper, we investigate a multivariate heteroscedastic Bayesian hierarchical model in which an informative prior with equicorrelated means is assumed. We estimate the mean vector by the shrinkage estimator based on Stein's unbiased risk estimation (SURE). It is shown that the squared error loss of the SURE estimator is close to that of an oracle estimator as the number of means grows. Our SURE estimator includes the SURE estimator under independence considered by Xie et al. (2012) as a special case. The finite-sample performance of our estimator is explored via simulations and two real data sets are used for illustration purposes.