SURE Estimates for a Heteroscedastic Hierarchical Model

• Published in: Journal of the American Statistical Association Vol. 107; no. 500; pp. 1465 - 1479
• Main Authors: Xie, Xianchao, Kou, S. C., Brown, Lawrence D.
• Format: Journal Article
• Language: English
• Published: United States Taylor & Francis Group 01.12.2012; Taylor & Francis Ltd
• Subjects: Asymptotic methods; Asymptotic optimality; Bayes estimators; Datasets; Estimating techniques; Estimation; Estimation methods; Estimators; Heteroscedasticity; Hierarchical scales; Maximum likelihood estimation; Oracles; Risk theory; RSS technology; Sample size; Sampling; Scientific method; Shrinkage; Shrinkage estimator; Statistical analysis; Statistical discrepancies; Statistical methods; Statistical models; Statistics; Stein's unbiased risk estimate (SURE); Theory and Methods; Unbiased estimators; asymptotic optimality; hierarchical model; shrinkage estimator; Stein’s unbiased risk estimate (SURE); Heteroscedasticity
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.2012.728154

Hierarchical models are extensively studied and widely used in statistics and many other scientific areas. They provide an effective tool for combining information from similar resources and achieving partial pooling of inference. Since the seminal work by James and Stein ( 1961 ) and Stein ( 1962 ), shrinkage estimation has become one major focus for hierarchical models. For the homoscedastic normal model, it is well known that shrinkage estimators, especially the James-Stein estimator, have good risk properties. The heteroscedastic model, though more appropriate for practical applications, is less well studied, and it is unclear what types of shrinkage estimators are superior in terms of the risk. We propose in this article a class of shrinkage estimators based on Stein's unbiased estimate of risk (SURE). We study asymptotic properties of various common estimators as the number of means to be estimated grows (p → ∞). We establish the asymptotic optimality property for the SURE estimators. We then extend our construction to create a class of semiparametric shrinkage estimators and establish corresponding asymptotic optimality results. We emphasize that though the form of our SURE estimators is partially obtained through a normal model at the sampling level, their optimality properties do not heavily depend on such distributional assumptions. We apply the methods to two real datasets and obtain encouraging results.