Multidimensional statistics, especially in the case where the sample size and the number of estimated parameters (or hypotheses tested) are comparable, is a natural area of application of the theory of random matrices (see V. L. Girko, Statistical Analysis of Observations of Increasing Dimension (Theory and Decision Library B). Springer, New York, 1995.). In this paper we consider procedures for multiple hypothesis testing that may be used in the case, which is even further from the classical statistics: the number of hypotheses may be much larger than the sample size. We consider general step-down multiple testing procedures – stopping-time-like decision rules that reject hypotheses sequentially, starting from those with the smallest (most significant) p -values. The most frequently used subclass of step-down procedures is that of threshold step-down (TSD) procedures exemplified by the classical Holm procedure. We obtain a formula that, for any step-down procedure satisfying a natural condition of monotonicity , provides the exact level at which the procedure controls the family-wise error rate – the probability of one or more false rejections. No assumptions are made about the joint distribution of p -values. The quantity is given by an easily computable expression. The result may be useful in all applications, where no parametric or other assumptions about the joint distribution of p -values can be legitimately made.
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