We congratulate Romano, Shaikh, and Wolf for their interesting work. Our only criticism to the presentation of the article, which is otherwise very readable, concerns Remark 1 on p. 8. This is crucial to understanding the method, because it explains that the estimates of the probabilities under the null are determined by the smaller test statistics, so it should have been made explicit at an earlier stage in Sect. 5. Incidentally, the use of ‘rth largest ’ and ‘rth smallest ’ to denote the rth order statistic on pp. 6 and 8 is confusing. The assumption that n is large and that the θj ’s are uniformly away from zero ensures that few non-null statistics will be mixed with the null ones and hence that the estimates of the probabilities in (10) are approximately correct. Since the models used in the simulation study conform to this assumption, we guess that the bootstrap method is shown here at its best. We wonder how it will perform under a sequence of alternatives which approach the null in a more continuous fashion, a more plausible scenario in real-life applications. One interesting aspect of the simulation results presented in Tables 1 and 2 is how well the ‘standard ’ Benjamini–Hochberg method (BH) works in all scenarios of dependence: the FDR is kept below the required 10%, while the power is on average 80 % of that of the bootstrap method proposed by the authors. This suggest
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