Exact calculations for false discovery proportion with application to least favorable configurations

Abstract : 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 {$s$-th} 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. This new approach is 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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Journal articles
The Annals of Statistics, IMS, 2011, 39 (1), pp.584-612. <10.1214/10-AOS847>
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Etienne Roquain, Fanny Villers. Exact calculations for false discovery proportion with application to least favorable configurations. The Annals of Statistics, IMS, 2011, 39 (1), pp.584-612. <10.1214/10-AOS847>. <hal-00456385v2>

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