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

TL;DR

This paper introduces a nonparametric empirical Bayes estimator for high-dimensional normal mean vectors that adapts well to sparsity and asymptotically approaches the optimal Bayes risk, with strong empirical performance.

Contribution

It proposes a computationally simple estimator that asymptotically matches the Bayes risk for unknown prior distributions in high-dimensional normal mean estimation.

Findings

Estimator performs well in simulations, especially in sparse settings.

Method adapts effectively to both sparse and nonsparse scenarios.

Asymptotic risk approaches the Bayes risk under broad conditions.

Abstract

We consider the classical problem of estimating a vector $\bolds{\mu}=(\mu_1,...,\mu_n)$ 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 $\hat{\bolds{\mu}}$, such that the ratio of its risk $E(\hat{\bolds{\mu}}-\bolds{\mu})^2$ 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 $\bolds{\mu}$ 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.