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Closure properties of classes of multiple testing procedures

机译：多个测试程序的类的封闭属性

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Statistical discoveries are often obtained through multiple hypothesis testing. A variety of procedures exists to evaluate multiple hypotheses, for instance the ones of Benjamini-Hochberg, Bonferroni, Holm or Sidak. We are particularly interested in multiple testing procedures with two desired properties: (solely) monotonic and well-behaved procedures. This article investigates to which extent the classes of (monotonic or well-behaved) multiple testing procedures, in particular the subclasses of so-called step-up and step-down procedures, are closed under basic set operations, specifically the union, intersection, difference and the complement of sets of rejected or non-rejected hypotheses. The present article proves two main results: First, taking the union or intersection of arbitrary (monotonic or well-behaved) multiple testing procedures results in new procedures which are monotonic but not well-behaved, whereas the complement or difference generally preserves neither property. Second, the two classes of (solely monotonic or well-behaved) step-up and step-down procedures are closed under taking the union or intersection, but not the complement or difference.
机译：统计发现通常是通过多重假设检验获得的。存在多种评估多个假设的程序，例如Benjamini-Hochberg，Bonferroni，Holm或Sidak。我们对具有两个所需属性的多种测试过程特别感兴趣：（仅）单调和行为良好的过程。本文研究了在基本设置操作（特别是并集，交集）下，（单调或行为良好的）多个测试过程的类别（尤其是所谓的逐步升级和逐步降低程序的子类）在何种程度上被关闭差异和一组被拒绝或未被拒绝的假设的补充。本文证明了两个主要结果：首先，采用任意（单调或行为良好）的多个测试过程的并集或交集会导致新的过程具有单调性，但表现不佳，而互补或差异通常都不会保留任何属性。其次，在采用并集或相交的情况下关闭（仅单调或行为良好的）两类升序和降级过程，但不求补或差。

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