ScholarlyCommons

Title

Nonparametric Empirical Bayes and Compound Decision Approaches to Estimation of a High-Dimensional Vector of Normal Means

Author(s)

Lawrence D. Brown, University of Pennsylvania
Eitan Greenshtein

Document Type

Journal Article

Date of this Version

2009

Publication Source

Annals of Statistics

Volume

37

Issue

4

Start Page

1685

Last Page

1704

DOI

10.1214/08-AOS630

Abstract

We consider the classical problem of estimating a vector μ=(μ₁, …, μ_n) based on independent observations Y_i∼N(μ_i, 1), i=1, …, n.

Suppose μ_i, 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(μ̂−μ)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.

Keywords

empirical Bayes, compound decision

Recommended Citation

Brown, L. D., & Greenshtein, E. (2009). Nonparametric Empirical Bayes and Compound Decision Approaches to Estimation of a High-Dimensional Vector of Normal Means. Annals of Statistics, 37 (4), 1685-1704. http://dx.doi.org/10.1214/08-AOS630