We present a novel necessary and sufficient principle for multiple testing methods controlling an expected loss. This principle asserts that every such multiple testing method is a special case of a general closed testing procedure based on e-values. It generalizes the Closure Principle, known to underlie all methods controlling familywise error and tail probabilities of false discovery proportions, to a large class of error rates -- in particular to the false discovery rate (FDR). By writing existing methods as special cases of this procedure, we can achieve uniform improvements, as we demonstrate for the e-Benjamini-Hochberg and the Benjamini-Yekutieli procedures, and the self-consistent method of Su (2018). We also show that methods derived using our novel e-Closure Principle generally control their error rate not just for one rejected set, but simultaneously over many, allowing post hoc flexibility for the researcher. Moreover, we show that because all multiple testing methods for all error metrics are derived from the same procedure, researchers may even choose the error metric post hoc. Under certain conditions, this flexibility even extends to post hoc choice of the nominal error rate.

翻译：我们提出了一种新颖的、控制期望损失的多重假设检验方法的必要且充分原理。该原理断言，所有此类多重假设检验方法都是基于e值的一般闭合检验程序的特例。它将已知为控制族错误率和错误发现比例尾部概率的所有方法之基础的闭合原理，推广至一大类错误率——特别是错误发现率（FDR）。通过将现有方法表述为此程序的特例，我们可以实现统一的改进，正如我们在e-Benjamini-Hochberg程序、Benjamini-Yekutieli程序以及Su（2018）的自洽方法中所展示的那样。我们还表明，使用我们新颖的e-闭合原理推导出的方法，通常不仅对一个被拒绝的集合控制其错误率，而且能同时对多个集合进行控制，从而为研究者提供了事后灵活性。此外，我们证明，由于所有针对所有误差度量的多重假设检验方法都源自同一程序，研究者甚至可以在事后选择误差度量。在某些条件下，这种灵活性甚至可以扩展到对名义错误率的事后选择。