Electronic Journal of Statistics

Variability and stability of the false discovery proportion

Marc Ditzhaus and Arnold Janssen

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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” $\text{FDP}=V/R$ 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.

Article information

Electron. J. Statist., Volume 13, Number 1 (2019), 882-910.

Received: August 2018
First available in Project Euclid: 26 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

False discovery rate (FDR) moments of the false discovery proportion (FDP) multiple testing adaptive Benjamini Hochberg methods Storey estimator $p$-values

Creative Commons Attribution 4.0 International License.


Ditzhaus, Marc; Janssen, Arnold. Variability and stability of the false discovery proportion. Electron. J. Statist. 13 (2019), no. 1, 882--910. doi:10.1214/19-EJS1544. https://projecteuclid.org/euclid.ejs/1553565707

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