We investigate asymptotically optimal multiple testing procedures for streams of sequential data in the context of prior information on the number of false null hypotheses ("signals"). We show that the "gap" and "gap-intersection" procedures, recently proposed and shown by Song and Fellouris (2017, Electron. J. Statist.) to be asymptotically optimal for controlling type 1 and 2 familywise error rates (FWEs), are also asymptotically optimal for controlling FDR/FNR when their critical values are appropriately adjusted. Generalizing this result, we show that these procedures, again with appropriately adjusted critical values, are asymptotically optimal for controlling any multiple testing error metric that is bounded between multiples of FWE in a certain sense. This class of metrics includes FDR/FNR but also pFDR/pFNR, the per-comparison and per-family error rates, and the false positive rate. Our analysis includes asymptotic regimes in which the number of null hypotheses approaches $\infty$ as the type 1 and 2 error metrics approach $0$.

中文翻译：

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具有（或不具有）关于信号数量的先验信息的渐近最优顺序 FDR 和 pFDR 控制

我们在关于假零假设（“信号”）数量的先验信息的背景下，研究了顺序数据流的渐近最优多重测试程序。我们表明，最近由 Song 和 Fellouris (2017, Electron. J. Statist.) 提出并展示的“间隙”和“间隙相交”程序对于控制 1 类和 2 类家庭错误率 (FWE) 是渐近最优的，当它们的临界值被适当调整时，它们对于控制 FDR/FNR 也是渐近最优的。概括这个结果，我们表明这些程序，同样具有适当调整的临界值，对于控制在某种意义上在 FWE 的倍数之间有界的任何多重测试误差度量是渐近最优的。此类指标包括 FDR/FNR 和 pFDR/pFNR，每个比较和每个家庭的错误率，以及误报率。我们的分析包括渐近机制，其中当类型 1 和 2 错误度量接近 $0$ 时，零假设的数量接近 $\infty$。