Compound Selection Decisions: An Almost SURE Approach

This paper proposes methods for producing compound selection decisions in a Gaussian sequence model. Given unknown, fixed parameters μ_ 1:n and known σ1:n with observations Yi ~ textsfN(μi, σi2), the decision maker would like to select a subset of indices S so as to maximize utility frac1nsumi∈ S (μi - Ki), for known costs Ki. Inspired by Stein's unbiased risk estimate (SURE), we introduce an almost unbiased estimator, called ASSURE, for the expected utility of a proposed decision rule. ASSURE allows a user to choose a welfare-maximizing rule from a pre-specified class by optimizing the estimated welfare, thereby producing selection decisions that borrow strength across noisy estimates. We show that ASSURE produces decision rules that are asymptotically no worse than the optimal but infeasible decision rule in the pre-specified class. We apply ASSURE to the selection of Census tracts for economic opportunity, the identification of discriminating firms, and the analysis of p-value decision procedures in A/B testing.