On SURE estimates in hierarchical models assuming heteroscedasticity for both levels of a two-level normal hierarchical model

In this paper, we consider the estimation problem of a set of normal population means with in the presence of heteroscedasticity in both levels of a two-level hierarchical model. We obtain weighted shrinkage estimates of population means based on weighted Stein’s unbiased risk estimate. This is achieved by first estimating the nuisance parameters of variances and then using them in shrinkage estimators of means. Our approach is different from the SURE estimators for heteroscedastic hierarchical model studied by Xie et al. (2012) in that they assume heteroscedasticity only in the response variable while we assume it in both levels. The asymptotic optimality properties of weighted shrinkage estimators are derived. It may happen that the variances of two sub-models are proportional, in this case one finds closed form formulas for the population mean estimators. Finally, we compare the performance of competing weighted shrinkage estimators in the context of analyzing two real data sets, showing our estimator outperforms the competitor SURE estimator.