Group-Linear Empirical Bayes Estimates for a Heteroscedastic Normal Mean

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...

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Bibliographic Details
Published in:Journal of the American Statistical Association Vol. 113; no. 522; pp. 698 - 710
Main Authors: Weinstein, Asaf, Ma, Zhuang, Brown, Lawrence D., Zhang, Cun-Hui
Format: Journal Article
Language:English
Published: Alexandria Taylor & Francis 03.04.2018
Taylor & Francis Group,LLC
Taylor & Francis Ltd
Subjects:
Adequacy
Ambition
Asymptotic optimality
Baseball
Compound decision
data collection
Empirical analysis
Empirical Bayes
equations
Estimates
Estimators
Groups
Heteroscedasticity
heteroskedasticity
Minimax technique
prediction
Quantitative analysis
Regression analysis
risk
Shrinkage estimator
Statistical methods
Statistics
Symmetry
Theory and Methods
ISSN:0162-1459, 1537-274X, 1537-274X
Online Access:Get full text
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Summary: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.
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ISSN:0162-1459
1537-274X
1537-274X
DOI:10.1080/01621459.2017.1280406