Journal
ANNALS OF STATISTICS
Volume 39, Issue 1, Pages 584-612Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/10-AOS847
Keywords
False discovery rate; false discovery proportion; multiple testing; least favorable configuration; power; equicorrelated multivariate normal distribution; step-up; step-down
Categories
Funding
- French Agence Nationale de la Recherche (ANR) [ANR-09-JCJC-0027-01, ANR-PARCIMONIE, ANR-09-JCJC-0101-01]
- French ministry of foreign and european affairs [21887 NJ]
- Agence Nationale de la Recherche (ANR) [ANR-09-JCJC-0101] Funding Source: Agence Nationale de la Recherche (ANR)
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In a context of multiple hypothesis testing, we provide several new exact calculations related to the false discovery proportion (FDP) of step-up and step-down procedures. For step-up procedures, we show that the number of erroneous rejections conditionally on the rejection number is simply a binomial variable, which leads to explicit computations of the c.d.f., the sth moment and the mean of the FDP, the latter corresponding to the false discovery rate (FDR). For step-down procedures, we derive what is to our knowledge the first explicit formula for the FDR valid for any alternative c.d.f. of the p-values. We also derive explicit computations of the power for both step-up and step-down procedures. These formulas are explicit in the sense that they only involve the parameters of the model and the c.d.f. of the order statistics of i.i.d. uniform variables. The p-values are assumed either independent or coming from an equicorrelated multivariate normal model and an additional mixture model for the true/false hypotheses is used. Our approach is then used to investigate new results which are of interest in their own right, related to least/most favorable configurations for the FDR and the variance of the FDP.
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