An important limitation of standard multiple testing procedures is that the null distribution should be known. Here, we consider a null distribution-free approach for multiple testing in the following semi-supervised setting: the user does not know the null distribution, but has at hand a single sample drawn from this null distribution. In practical situations, this null training sample (NTS) can come from previous experiments, from a part of the data under test, from specific simulations, or from a sampling process. In this work, we present theoretical results that handle such a framework, with a focus on the false discovery rate (FDR) control and the Benjamini-Hochberg (BH) procedure. First, we introduce a procedure providing strong FDR control. Second, we also give a power analysis for that procedure suggesting that the price to pay for ignoring the null distribution is low when the NTS sample size $n$ is sufficiently large in front of the number of test $m$; namely $n\gtrsim m/(\max(1,k))$, where $k$ denotes the number of "detectable" alternatives. Third, to complete the picture, we also present a negative result that evidences an intrinsic transition phase to the general semi-supervised multiple testing problem {and shows that the proposed method is optimal in the sense that its performance boundary follows this transition phase}. Our theoretical properties are supported by numerical experiments, which also show that the delineated boundary is of correct order without further tuning any constant. Finally, we demonstrate that our approach provides a theoretical ground for standard practice in astronomical data analysis, and in particular for the procedure proposed in \cite{Origin2020} for galaxy detection.
翻译:标准多重测试程序的一个重要限制是应该知道无效分布 。 这里, 我们考虑在以下半监督环境下对多重测试采用无效分配方法 : 用户不知道无效分布, 但手头有一个从无效分布中提取的单一样本 。 在实际情况下, 这种无效培训样本可以来自先前的实验, 来自测试中数据的一部分, 来自特定模拟, 或来自抽样过程 。 在这项工作中, 我们提出处理这样一个框架的理论结果, 重点是虚假的发现率( FDR) 控制和 Benjami- Hochberg (BH) 程序。 首先, 我们引入了一个程序, 提供强大的 FDR 控制。 其次, 我们还对这个程序进行权力分析, 表明当NTS 样本规模的美元在测试数量之前足够大的时候, 也可以来自一个测试 $\ gtrsimm/ (\max(1, k) 美元), 其中, 以 $k 表示“ 解析” 和 Benjaniversalal 选项的过渡过程, 也显示一个我们提出的连续测试阶段的模拟过程 。