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 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 tested). 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 arguably amenable to extensions and generalizations. We end with a few pressing open questions that we think our paper opens a door to solving.

翻译：多重检验文献主要处理p值之间的三种相依性假设：独立性、正回归相依性和任意相依性。本文针对各种负相依概念（负高斯相依、负关联、负象限相依及弱负相依）提供了我们认为的首批理论结果。这些结果涵盖Simes全局零假设检验和Benjamini-Hochberg程序——实验表明这两种方法在负相依条件下存在保守性不足的问题。这两种方法的保守性不足程度被限制在低于任意相依条件（特别是与待检验假设数量无关的因子）的范围内。我们同时给出了关于负相依e值的新结果，并提供了负相依可能出现的若干实例。本文证明方法基础且简洁，因而具有扩展推广的可行性。最后，我们提出了若干亟待解决的关键问题，认为本文为这些问题开辟了解决途径。