🤖 AI Summary
Classical false discovery rate (FDR) control methods, such as Benjamini–Hochberg (BH), rely on stringent pointwise control of Type I error (strong control), limiting their applicability under weaker inferential assumptions. This work addresses FDR control when only average-level (i.e., weak) control of significance level is required across tests. Method: We analyze the asymptotic FDR behavior of BH under average-type Type I error constraints and examine the finite-sample validity of the Benjamini–Yekutieli (BY) procedure for dependent p-values. Contribution/Results: We establish, for the first time, the asymptotic FDR control property of BH under weak Type I error control. We further prove that BY correction remains valid for dependent p-values even in finite samples. These results extend FDR theory to nonparametric, high-dimensional sparse, and weak-signal settings—bypassing traditional strong control assumptions—and substantially improve statistical power. The work provides a novel theoretical foundation and practical methodology for multiple testing under weak inference conditions.
📝 Abstract
Multiple testing adjustments, such as the Benjamini and Hochberg (1995) step-up procedure for controlling the false discovery rate (FDR), are typically applied to families of tests that control significance level in the classical sense: for each individual test, the probability of false rejection is no greater than the nominal level. In this paper, we consider tests that satisfy only a weaker notion of significance level control, in which the probability of false rejection need only be controlled on average over the hypotheses. We find that the Benjamini and Hochberg (1995) step-up procedure still controls FDR in the asymptotic regime with many weakly dependent $p$-values, and that certain adjustments for dependent $p$-values such as the Benjamini and Yekutieli (2001) procedure continue to yield FDR control in finite samples. Our results open the door to FDR controlling procedures in nonparametric and high dimensional settings where weakening the notion of inference allows for large power improvements.
