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 .=(.1, ., .n) based on independent observations Yi.N(.i, 1), i=1, ., n. Suppose .i, i=1, ., n are independent realizations from a completely unknown G. We suggest an easily computed estimator .., such that the ratio of its risk E(....)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 . 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.