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

We consider the classical problem of estimating a vector μ=(μ1, …, μn) based on independent observations Yi∼N(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(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.