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Multiple testing adjustments, such as the Benjamini and Hochberg (1995)step-up procedure for controlling the false discovery rate (FDR), are typicallyapplied to families of tests that control significance level in the classicalsense: for each individual test, the probability of false rejection is nogreater than the nominal level. In this paper, we consider tests that satisfyonly a weaker notion of significance level control, in which the probability offalse rejection need only be controlled on average over the hypotheses. We findthat the Benjamini and Hochberg (1995) step-up procedure still controls FDR inthe asymptotic regime with many weakly dependent $p$-values, and that certainadjustments for dependent $p$-values such as the Benjamini and Yekutieli (2001)procedure continue to yield FDR control in finite samples. Our results open thedoor to FDR controlling procedures in nonparametric and high dimensionalsettings where weakening the notion of inference allows for large powerimprovements.

False Discovery Rate Adjustments for Average Significance Level Controlling Tests