Variance estimation in compound decision theory under boundedness

The normal means model is often studied under the assumption of a known variance. However, ignorance of the variance is a frequent issue in applications and basic theoretical questions still remain open in this setting. This article establishes that the sharp minimax rate of variance estimation in square error is (log log n/ log n ) 2 under arguably the most mild assumption imposed for identifiability: bounded means. The rate-optimal estimator proposed in this article achieves the optimal rate by estimating O (log n/ log log n ) cumulants and leveraging a variational representation of the noise variance in terms of the cumulants of the data distribution. The minimax lower bound involves a moment matching construction