Intrinsic Bounds and False Discovery Rate Control in Multiple Testing Problems

Details Intrinsic Bounds and False Discovery Rate Control in Multiple Testing Problems Intrinsic Bounds and False Discovery Rate Control in Multiple Testing Problems

2010-03-03

arxiv.org/abs/1003.0747v1

Mathematics

Statistics Theory

Scientific paper

When testing a large number of independent hypotheses, three different questions are of interest: are some hypotheses true alternatives? How many of them? Which of them? These questions give rise to a detection, an estimation, and a selection problem. Recent work demonstrates the existence of intrinsic bounds in these problems: detection and estimation boundaries in sparse location models, and criticality for the selection problem. We study consequences of such limitations in terms of power of False Discovery Rate (FDR) controlling procedures. FDR is the expected False Discovery Proportion (FDP), that is, the expected proportion of false rejections among all rejected hypotheses. For the selection problem, we illustrate the connection between criticality and the regularity of the distribution of the test statistics, and discuss expected and observed consequences of criticality in terms of power of FDR controlling procedures, on both simulated and real data. For the problem of estimating the fraction of true null hypotheses, we make explicit connections between the parameters of the multiple testing problem and consistency and convergence rates of a broad class of non-parametric estimators, and prove that these convergence rates determine that of the FDP achieved by "plug-in" multiple testing procedures, which are incorporating such an estimator in order to yield tighter FDR control.

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