大规模多重检验：错误发现率控制的基本极限与复合 Oracle
Large-scale Multiple Testing: Fundamental Limits of False Discovery Rate Control and Compound Oracle
AI总结：
本文研究大规模多重检验中 FDR 与 FNR 的根本权衡，证明最优 FDR–FNR 边界需要复合规则，并给出高概率 FDP 控制下的对应最优结果。
摘要：
错误发现率（FDR）和错误未发现率（FNR）被定义为错误发现比例（FDP）和错误未发现比例（FNP）的期望，是多重检验中最常用的基准。尽管近年来理论和算法均有进展，但除了某些受限的决策规则类别（例如可分规则）或其他性能指标（例如边际 FDR 与边际 FNR，即 mFDR 和 mFNR）之外，FDR 与 FNR 之间的最优权衡在很大程度上仍不明确。本文中，我们在假设数量趋于无穷时，确定了两组分随机混合模型下渐近最优的 FDR–FNR 权衡。与由可分决策规则实现的最优 mFDR–mFNR 权衡不同，最优 FDR–FNR 权衡即使在大样本极限下、且对于高斯位置模型这样简单的模型，也需要复合规则。可分规则的这种次优性也适用于其他目标，例如最大化真实发现的期望数量。最后，为解决 FDR 只控制 FDP 的期望而不控制其波动这一局限，我们还确定了以高概率控制 FDP 时的最优权衡，并表明它与 mFDR 和 mFNR 的最优权衡一致。文中还得到了向具有固定非零假设比例模型的扩展。
英文摘要：
The false discovery rate (FDR) and the false non-discovery rate (FNR), defined as the expected false discovery proportion (FDP) and the false non-discovery proportion (FNP), are the most popular benchmarks for multiple testing. Despite the theoretical and algorithmic advances in recent years, the optimal tradeoff between the FDR and the FNR has been largely unknown except for certain restricted classes of decision rules, e.g., separable rules, or for other performance metrics, e.g., the marginal FDR and the marginal FNR (mFDR and mFNR). In this paper, we determine the asymptotically optimal FDR-FNR tradeoff under the two-group random mixture model when the number of hypotheses tends to infinity. Distinct from the optimal mFDR-mFNR tradeoff, which is achieved by separable decision rules, the optimal FDR-FNR tradeoff requires compound rules even in the large-sample limit and for models as simple as the Gaussian location model. This suboptimality of separable rules also holds for other objectives, such as maximizing the expected number of true discoveries. Finally, to address the limitation of the FDR which only controls the expectation but not the fluctuation of the FDP, we also determine the optimal tradeoff when the FDP is controlled with high probability and show it coincides with that of the mFDR and the mFNR. Extensions to models with a fixed non-null proportion are also obtained.
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