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Optimality of the Holm procedure among general step-down multiple testing procedures

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  • Gordon, Alexander Y.
  • Salzman, Peter

Abstract

We study the class of general step-down multiple testing procedures, which contains the usually considered procedures determined by a nondecreasing sequence of thresholds (we call them threshold step-down, or TSD, procedures) as a parametric subclass. We show that all procedures in this class satisfying the natural condition of monotonicity and controlling the family-wise error rate (FWER) at a prescribed level are dominated by one of them -- the classical Holm procedure. This generalizes an earlier result pertaining to the subclass of TSD procedures [Lehmann, E.L., Romano, J.P., 2005. Testing Statistical Hypotheses, 3rd ed. Springer, New York]. We also derive a relation between the levels at which a monotone step-down procedure controls the FWER and the generalized FWER (the probability of k or more false rejections).

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  • Gordon, Alexander Y. & Salzman, Peter, 2008. "Optimality of the Holm procedure among general step-down multiple testing procedures," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1878-1884, September.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:13:p:1878-1884
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    References listed on IDEAS

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    1. Gordon, Alexander Y., 2007. "Unimprovability of the Bonferroni procedure in the class of general step-up multiple testing procedures," Statistics & Probability Letters, Elsevier, vol. 77(2), pages 117-122, January.
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    Cited by:

    1. Pallavi Basu & Luella Fu & Alessio Saretto & Wenguang Sun, 2021. "Empirical Bayes Control of the False Discovery Exceedance," Working Papers 2115, Federal Reserve Bank of Dallas.
    2. Gordon, Alexander Y., 2009. "Inequalities between generalized familywise error rates of a multiple testing procedure," Statistics & Probability Letters, Elsevier, vol. 79(19), pages 1996-2004, October.
    3. Gordon, Alexander Y., 2012. "A sharp upper bound for the expected number of false rejections," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1507-1514.
    4. Georg Hahn, 2018. "Closure properties of classes of multiple testing procedures," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 102(2), pages 167-178, April.

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    1. Georg Hahn, 2018. "Closure properties of classes of multiple testing procedures," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 102(2), pages 167-178, April.
    2. Gordon, Alexander Y., 2014. "Smoothing of stepwise multiple testing procedures," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 149-157.
    3. Gordon, Alexander Y., 2009. "Inequalities between generalized familywise error rates of a multiple testing procedure," Statistics & Probability Letters, Elsevier, vol. 79(19), pages 1996-2004, October.
    4. Gordon, Alexander Y., 2012. "A sharp upper bound for the expected number of false rejections," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1507-1514.

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