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

Abstract

We consider the classical problem of estimating a vector $\boldsμ=(μ_1,...,μ_n)$ based on independent observations $Y_i\sim N({\mu }_i,1)$, $i=1,...,n$. Suppose ${\mu }_i$, $i=1,...,n$ are independent realizations from a completely unknown $G$. We suggest an easily computed estimator $\hat{\boldsμ}$, such that the ratio of its risk $E(\hat{\boldsμ}-\boldsμ)^2$ 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 $\boldsμ$ 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.

Published in at http://dx.doi.org/10.1214/08-AOS630 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences
• General maximum likelihood empirical Bayes estimation of normal means
• In-season prediction of batting averages: A field test of empirical Bayes and Bayes methodologies
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