## Asymptotic Results on Adaptive False Discovery Rate Controlling Procedures Based on Kernel Estimators

*Pierre Neuvial*; 14(May):1423−1459, 2013.

### Abstract

The False Discovery Rate (FDR) is a commonly used type I error
rate in multiple testing problems. It is defined as the expected
False Discovery Proportion (FDP), that is, the expected fraction
of false positives among rejected hypotheses. When the
hypotheses are independent, the Benjamini-Hochberg procedure
achieves FDR control at any pre-specified level. By
construction, FDR control offers no guarantee in terms of power,
or type II error. A number of alternative procedures have been
developed, including plug-in procedures that aim at gaining
power by incorporating an estimate of the proportion of true
null hypotheses. In this paper, we study the asymptotic behavior
of a class of plug-in procedures based on kernel estimators of
the density of the $p$-values, as the number $m$ of tested
hypotheses grows to infinity. In a setting where the hypotheses
tested are independent, we prove that these procedures are
asymptotically more powerful in two respects: (i) a tighter
asymptotic FDR control for any target FDR level and (ii) a
broader range of target levels yielding positive asymptotic
power. We also show that this increased asymptotic power comes
at the price of slower, non-parametric convergence rates for the
FDP. These rates are of the form $m^{-k/(2k+1)}$, where $k$ is
determined by the regularity of the density of the $p$-value
distribution, or, equivalently, of the test statistics
distribution. These results are applied to one- and two-sided
tests statistics for Gaussian and Laplace location models, and
for the Student model.

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