Abstract
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” $\text{FDP}=V/R$ 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 stability of the FDP, i.e., when the FDP is asymptotically close to its mean. We present sufficient and necessary conditions for this stability. They include the presence of a stable estimator for the proportion $m_{0}/m$. 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 them.
Highlights
Testing m ≥ 2 hypotheses simultaneously is a frequent issue in statistical practice, e.g., in genomic research
The false discovery rate” (FDR) is the expectation of the “false discovery proportion” (FDP), the ratio
Under the so-called basic independence (BI) assumption, which will be introduced in more detail below, the classical Benjamini and Hochberg linear step-up test, in short BH test, is α-controlling
Summary
Testing m ≥ 2 hypotheses simultaneously is a frequent issue in statistical practice, e.g., in genomic research. Generalized Storey estimators with data dependent weights discussed by Heesen and Janssen [24] will be our prime example in later discussions of our general results The latter lead to α-controlling procedures, whereas other approaches are often just asymptotically α-controlling, i.e., lim supm→∞ FDRm ≤ α. Ferreira and Zwinderman [15] presented formulas for higher moments of FDPm for the BH test and Roquain and Villers [31] did so for step-up and step-down tests with general (but data independent) critical values We extend these formulas to adaptive procedures.
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