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 $Y_i\sim N(μ_i,1)$, $i=1,...,n$. Suppose $μ_i$, $i=1,...,n$ are independent realizations from a completely unknown $G$. We suggest an easily computed estimator $\hat{\boldsμ}$, 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.