Group-Linear Empirical Bayes Estimates for a Heteroscedastic Normal Mean

• 成果类型: Article
• 署名作者: Weinstein, Asaf; Ma, Zhuang; Brown, Lawrence D.; Zhang, Cun-Hui
• 署名单位: Stanford University; University of Pennsylvania; University of Pennsylvania; Rutgers University System; Rutgers University New Brunswick
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2017.1280406
• 发表日期: 2018
• 页码: 698-710
• 关键词: minimax estimation

摘要

The problem of estimating the mean of a normal vector with known but unequal variances introduces substantial difficulties that impair the adequacy of traditional empirical Bayes estimators. By taking a different approach that treats the known variances as part of the random observations, we restore symmetry and thus the effectiveness of such methods. We suggest a group-linear empirical Bayes estimator, which collects observations with similar variances and applies a spherically symmetric estimator to each group separately. The proposed estimator is motivated by a new oracle rule which is stronger than the best linear rule, and thus provides a more ambitious benchmark than that considered in the previous literature. Our estimator asymptotically achieves the new oracle risk (under appropriate conditions) and at the same time is minimax. The group-linear estimator is particularly advantageous in situations where the true means and observed variances are empirically dependent. To demonstrate the merits of the proposed methods in real applications, we analyze the baseball data used by Brown (2008), where the group-linear methods achieved the prediction error of the best nonparametric estimates that have been applied to the dataset, and significantly lower error than other parametric and semiparametric empirical Bayes estimators.