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We study empirical Bayes estimation of the effect sizes of $N$ units from $K$noisy observations on each unit. We show that it is possible to achievenear-Bayes optimal mean squared error, without any assumptions or knowledgeabout the effect size distribution or the noise. The noise distribution can beheteroskedastic and vary arbitrarily from unit to unit. Our proposal, which wecall Aurora, leverages the replication inherent in the $K$ observations perunit and recasts the effect size estimation problem as a general regressionproblem. Aurora with linear regression provably matches the performance of awide array of estimators including the sample mean, the trimmed mean, thesample median, as well as James-Stein shrunk versions thereof. Aurora automateseffect size estimation for Internet-scale datasets, as we demonstrate on datafrom a large technology firm.

Empirical Bayes mean estimation with nonparametric errors via order statistic regression on replicated data