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 class 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 and randomization even in the large-sample limit. A data-driven version of the oracle rule is proposed and shown to outperform existing methodologies on simulated data for models as simple as the normal mean model. Finally, to address the limitation of the FDR and FNR which only control the expectations but not the fluctuations of the FDP and FNP, we also determine the optimal tradeoff when the FDP and FNP are controlled with high probability and show it coincides with that of the mFDR and the mFNR.

翻译：错误发现率（FDR）与错误非发现率（FNR）分别定义为错误发现比例（FDP）与错误非发现比例（FNP）的期望，是多重检验中最常用的基准指标。尽管近年来在理论与算法方面取得了诸多进展，但FDR与FNR之间的最优权衡关系在很大程度上仍属未知——这仅限于某些特定类别的决策规则（如可分离规则）或其他性能指标（如边际FDR与边际FNR，即mFDR与mFNR）。本文在假设数量趋于无穷的条件下，确定了双组随机混合模型下FDR与FNR的渐近最优权衡关系。与可分离决策规则可实现的最优mFDR-mFNR权衡不同，最优FDR-FNR权衡即使在样本量趋于无穷时也需要复合规则与随机化。我们提出了一种数据驱动的Oracle规则，并证明对于像正态均值模型这样简单的模型，该方法在模拟数据上优于现有方法。最后，针对FDR与FNR仅控制FDP与FNP的期望而非其波动的局限性，我们进一步确定了在大概率控制FDP与FNP条件下的最优权衡关系，并证明其与mFDR及mFNR的最优权衡一致。