The multiple testing literature has primarily dealt with three types of dependence assumptions between p-values: independence, positive regression dependence, and arbitrary dependence. In this paper, we provide what we believe are the first theoretical results under various notions of negative dependence (negative Gaussian dependence, negative regression dependence, negative association, negative orthant dependence and weak negative dependence). These include the Simes global null test and the Benjamini-Hochberg procedure, which are known experimentally to be anti-conservative under negative dependence. The anti-conservativeness of these procedures is bounded by factors smaller than that under arbitrary dependence (in particular, by factors independent of the number of hypotheses). We also provide new results about negatively dependent e-values, and provide several examples as to when negative dependence may arise. Our proofs are elementary and short, thus amenable to extensions.

翻译：多重假设检验文献主要处理p值之间的三类相依性假设：独立性、正回归相依性及任意相依性。本文针对各类负相依概念（负高斯相依、负回归相依、负关联、负象限相依与弱负相依）提出了我们认为的首批理论结果。这些结果涵盖经典的Simes全局零假设检验与Benjamini-Hochberg过程——实验研究表明这些方法在负相依条件下具有反保守性。我们证明了这些过程的反保守程度受限于比任意相依情形更小的因子（尤其是与假设数量无关的因子）。本文还给出了负相依e值的新结果，并提供了若干可能产生负相依性的实例。我们的证明方法初等且简洁，便于后续扩展。