In a paper published in 1939 in The Annals of Mathematical Statistics, Wald and Wolfowitz discussed the possible validity of a probability inequality between one- and two-sided coverage probabilities for the empirical distribution function. Twenty-eight years later, Vandewiele and Noé proved this inequality for Kolmogorov-Smirnov type goodness of fit tests. We refer to this type of inequality as one-two inequality. In this paper, we generalize their result for one- and two-sided union-intersection tests based on positively associated random variables and processes. Thereby, we give a brief review of different notions of positive association and corresponding results. Moreover, we introduce the notion of one-two dependence and discuss relationships with other dependence concepts. While positive association implies one-two dependence, the reverse implication fails. Last but not least, the Bonferroni inequality and the one-two inequality yield lower and upper bounds for two-sided acceptance/rejection probabilities which differ only slightly for significance levels not too large. We discuss several examples where the one-two inequality applies. Finally, we briefly discuss the possible impact of the validity of a one-two inequality on directional error control in multiple testing.

中文翻译：

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单边联合交叉检验和双边联合交叉检验之间的一二相关性和概率不等式

在 1939 年发表在《数理统计年鉴》上的一篇论文中，Wald 和 Wolfowitz 讨论了经验分布函数的单边覆盖概率和双边覆盖概率之间的概率不等式的可能有效性。28 年后，Vandewiele 和 Noé 证明了 Kolmogorov-Smirnov 类型拟合优度检验的不等式。我们将这种不等式称为一二不等式。在本文中，我们将他们的结果推广到基于正相关随机变量和过程的单边和双边联合交叉检验。因此，我们简要回顾了正关联的不同概念和相应的结果。此外，我们引入了一二依赖的概念，并讨论了与其他依赖概念的关系。虽然正关联意味着一二依赖，但反向暗示却失败了。最后但并非最不重要的，Bonferroni 不等式和一二不等式产生双边接受/拒绝概率的下限和上限，对于不太大的显着性水平，它们仅略有不同。我们讨论了几个适用一二不等式的例子。最后，我们简要讨论一二不等式的有效性对多重测试中方向误差控制的可能影响。