Family wise separation rates for multiple testing
Résumé
Starting from a parallel between some minimax adaptive tests of a single null hypothesis, based on aggregation approaches, and some tests of multiple hypotheses, we propose a new second kind error-related evaluation criterion, as the core of an emergent minimax theory for multiple tests. Aggregation-based tests are justified through their first kind error rate, which is controlled by the prescribed level on the one hand, and through their separation rates over various classes of alternatives, rates that are minimax on the other hand. We show that these tests can be viewed as the first steps of classical step-down multiple testing procedures, and ac-cordingly be evaluated from the multiple testing point of view also, through a control of their Family-Wise Error Rate (FWER). Conversely, many multiple testing procedures, from the historical ones of Bonferroni and Holm, to more recent ones like min-p procedures or randomized procedures, can be investigated from the minimax adaptive testing point of view. To this end, we extend the notion of separation rate to the multiple testing field, by defining the weak Family-Wise Separation Rate and its stronger counterpart, the Family-Wise Separation Rate (FWSR). As for non-parametric tests of a single null hypothesis, we prove that these new concepts allow an accurate analysis of the second kind error of a multiple testing procedure, leading to clear definitions of minimax and minimax adaptive multiple tests. Some illustrations in a classical Gaussian framework corroborate several expected results under particular conditions on the tested hypotheses, but also lead to more surprising results.
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