2026/07/19 by Xianyang Zhang
#math.ST #stat.ME #stat.TH
Many multiple-testing methods compare the two sides of a null distribution to control the false discovery rate (FDR). The mirror and knockoff+ thresholds use large positive scores or small p-values as discoveries and the opposite tail as a control. For valid knockoff statistics, this comparison is justified because, conditional on their magnitudes and the nonnull information, the null signs are independent fair coin flips. We show what can go wrong without this property. First, we construct exactly uniform p-values satisfying positive regression dependence on a subset (PRDS), with a positive joint density on the whole unit cube. At nominal level 10%, an eleven-hypothesis example has FDR 17.4%, and the FDR can approach one half. Second, for standard Gaussian null scores with any fixed positive equicorrelation, the liminf of the FDR is at least one half. Third, we construct exchangeable, pairwise-uncorrelated scores with a common continuous symmetric distribution for which the FDR is arbitrarily close to one; transforming them by their common null distribution gives exactly uniform p-values with the same failure. Finally, for target levels q<1/2, no deterministic monotone threshold based only on the two current tail counts gives a nontrivial distribution-free repair over the full-support PRDS class: it either fails or never rejects. These results do not contradict knockoff theory. They show that adaptively comparing control-side and discovery-side counts is not, by itself, an FDR guarantee. Validity depends on the joint behavior of the null signs, not only on marginal symmetry, Gaussianity, PRDS, exchangeability, or pairwise uncorrelatedness.