On the consistency of adaptive multiple tests
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Much effort has been done to control the "false discovery rate" (FDR) when m hypotheses are tested simultaneously. The FDR is the expectation of the "false discovery proportion" FDP=V/R given by the ratio of the number of false rejections V and all rejections R. In this paper, we have a closer look at the FDP for adaptive linear step-up multiple tests. These tests extend the well known Benjamini and Hochberg test by estimating the unknown amount m_0 of the true null hypotheses. We give exact finite sample formulas for higher moments of the FDP and, in particular, for its variance. Using these allows us a precise discussion about the consistency of adaptive step-up tests. We present sufficient and necessary conditions for consistency on the estimators m_0 and the underlying probability regime. We apply our results to convex combinations of generalized Storey type estimators with various tuning parameters and (possibly) data-driven weights. The corresponding step-up tests allow a flexible adaptation. Moreover, these tests control the FDR at finite sample size. We compare these tests to the classical Benjamini and Hochberg test and discuss the advantages of it.
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