Asymptotically minimax empirical Bayes estimation of a sparse normal mean

We consider the classical and practically important problem of estimating a sparse high-dimensional normal mean vector. Our proposed empirical Bayes model features a unique form of shrinkage, and has strong finite-sample performance. We prove, under mild conditions, that our empirical Bayes posterior concentrates on balls, centered at the true mean, with squared radius proportional to the frequentist minimax rate for the given sparsity class. Asymptotic minimaxity of the corresponding empirical Bayes posterior mean is also shown. Simulation results demonstrate the strong finite-sample performance of the proposed method against a variety of popular alternatives.