The Annals of Statistics

Nonparametric empirical Bayes and compound decision approaches to estimation of a high-dimensional vector of normal means

Lawrence D. Brown and Eitan Greenshtein

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We consider the classical problem of estimating a vector μ=(μ1, …, μn) based on independent observations YiN(μi, 1), i=1, …, n.

Suppose μi, i=1, …, n are independent realizations from a completely unknown G. We suggest an easily computed estimator μ̂, such that the ratio of its risk E(μ̂μ)2 with that of the Bayes procedure approaches 1. A related compound decision result is also obtained.

Our asymptotics is of a triangular array; that is, we allow the distribution G to depend on n. Thus, our theoretical asymptotic results are also meaningful in situations where the vector μ is sparse and the proportion of zero coordinates approaches 1.

We demonstrate the performance of our estimator in simulations, emphasizing sparse setups. In “moderately-sparse” situations, our procedure performs very well compared to known procedures tailored for sparse setups. It also adapts well to nonsparse situations.

Article information

Ann. Statist., Volume 37, Number 4 (2009), 1685-1704.

First available in Project Euclid: 18 June 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C12: Empirical decision procedures; empirical Bayes procedures 62C25: Compound decision problems

Empirical Bayes compound decision


Brown, Lawrence D.; Greenshtein, Eitan. Nonparametric empirical Bayes and compound decision approaches to estimation of a high-dimensional vector of normal means. Ann. Statist. 37 (2009), no. 4, 1685--1704. doi:10.1214/08-AOS630.

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