On SURE-Type Double Shrinkage Estimation

ABSTRACT

The article is concerned with empirical Bayes shrinkage estimators for the heteroscedastic hierarchical normal model using Stein's unbiased estimate of risk (SURE). Recently, Xie, Kou, and Brown proposed a class of estimators for this type of problems and established their asymptotic optimality properties under the assumption of known but unequal variances. In this article, we consider this problem with unequal and unknown variances, which may be more appropriate in real situations. By placing priors for both means and variances, we propose novel SURE-type double shrinkage estimators that shrink both means and variances. Optimal properties for these estimators are derived under certain regularity conditions. Extensive simulation studies are conducted to compare the newly developed methods with other shrinkage techniques. Finally, the methods are applied to the well-known baseball dataset and a gene expression dataset. Supplementary materials for this article are available online.

KEYWORDS:

• Double shrinkage estimators
• Empirical Bayes
• Hierarchical model
• Stein's unbiased risk estimate (SURE)

Supplementary Materials

The supplementary materials provide detailed proofs of main results.

Acknowledgments

The authors thank the editor, an associate editor, and three anonymous referees for their very helpful and constructive comments and suggestions that have led to significant improvements in the article.

Funding

Jing's research was partially supported by Hong Kong RGC6022/13P and RGC6019/12P. Li's research was supported by the National Natural Science Foundation of China (11201207, 11571154) and in part by the Fundamental Research Funds for the Central Universities (lzujbky-2015-74). Pan's research was partially supported by a MOE Tier 2 grant 2014-T2-2-060 and a MOE Tier 1 Grant RG25/14 at the Nanyang Technological University, Singapore. Zhou's research was partially supported by a grant R-155-000-139-112 at the National University of Singapore.