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The false discovery rate (FDR) and the false non-discovery rate (FNR),defined as the expected false discovery proportion (FDP) and the falsenon-discovery proportion (FNP), are the most popular benchmarks for multipletesting. Despite the theoretical and algorithmic advances in recent years, theoptimal tradeoff between the FDR and the FNR has been largely unknown exceptfor certain restricted class of decision rules, e.g., separable rules, or forother performance metrics, e.g., the marginal FDR and the marginal FNR (mFDRand mFNR). In this paper we determine the asymptotically optimal FDR-FNRtradeoff under the two-group random mixture model when the number of hypothesestends to infinity. Distinct from the optimal mFDR-mFNR tradeoff, which isachieved by separable decision rules, the optimal FDR-FNR tradeoff requirescompound rules and randomization even in the large-sample limit. A data-drivenversion of the oracle rule is proposed and shown to outperform existingmethodologies 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 theexpectations but not the fluctuations of the FDP and FNP, we also determine theoptimal tradeoff when the FDP and FNP are controlled with high probability andshow it coincides with that of the mFDR and the mFNR.

Large-scale Multiple Testing: Fundamental Limits of False Discovery Rate Control and Compound Oracle