This paper proposes methods for producing compound selection decisions in a Gaussian sequence model. Given unknown, fixed parameters $μ_ {1:n}$ and known $σ_{1:n}$ with observations $Y_i \sim \textsf{N}(μ_i, σ_i^2)$, the decision maker would like to select a subset of indices $S$ so as to maximize utility $\frac{1}{n}\sum_{i\in S} (μ_i - K_i)$, for known costs $K_i$. 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.

翻译：本文提出在高斯序列模型中生成复合选择决策的方法。给定未知固定参数$μ_{1:n}$与已知方差$σ_{1:n}$，观测值$Y_i \sim \textsf{N}(μ_i, σ_i^2)$，决策者需选择指标子集$S$，以最大化效用函数$\frac{1}{n}\sum_{i\in S} (μ_i - K_i)$，其中$K_i$为已知成本。受Stein无偏风险估计(SURE)启发，我们引入一种称为ASSURE的近乎无偏估计量，用于评估决策规则的期望效用。ASSURE允许用户通过优化估计福利，从预设规则类中选择福利最大化规则，从而生成能够跨噪声估计借用统计强度的选择决策。我们证明ASSURE生成的决策规则在渐近意义上不劣于预设规则类中最优但不可实现的决策规则。我们将ASSURE应用于经济机会普查区选择、歧视性企业识别，以及A/B测试中$p$值决策程序的分析。