26.03.2026 (Thursday)
We propose methods for compound selection decisions in a Gaussian sequence model. Given unknown, fixed parameters $\mu_{1:n}$, known $\sigma_{1:n}$, observations $Y_i \sim \Norm(\mu_i, \sigma_i^2)$, and known costs $K_i$, the decision maker chooses a subset $S
\subset [n]$ to maximize utility $\frac{1}{n}\sum_{i\in S} (\mu_i - K_i)$. Inspired by Stein's unbiased risk estimate (SURE), we introduce an almost-unbiased estimator, ASSURE, for the expected utility of a proposed decision rule. ASSURE selects a welfare‑maximizing rule within a pre‑specified class by optimizing the estimated welfare, thereby borrowing strength across noisy estimates. We show that, within the pre-specified class, ASSURE's decisions are asymptotically no worse than the optimal (infeasible) rule. 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.