Mathematics – Statistics Theory
Scientific paper
2008-02-11
Electronic Journal of Statistics 2 (2008) 963-992
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/08-EJS180 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by t
Scientific paper
10.1214/08-EJS180
We show that the control of the false discovery rate (FDR) for a multiple testing procedure is implied by two coupled simple sufficient conditions. The first one, which we call ``self-consistency condition'', concerns the algorithm itself, and the second, called ``dependency control condition'' is related to the dependency assumptions on the $p$-value family. Many standard multiple testing procedures are self-consistent (e.g. step-up, step-down or step-up-down procedures), and we prove that the dependency control condition can be fulfilled when choosing correspondingly appropriate rejection functions, in three classical types of dependency: independence, positive dependency (PRDS) and unspecified dependency. As a consequence, we recover earlier results through simple and unifying proofs while extending their scope to several regards: weighted FDR, $p$-value reweighting, new family of step-up procedures under unspecified $p$-value dependency and adaptive step-up procedures. We give additional examples of other possible applications. This framework also allows for defining and studying FDR control for multiple testing procedures over a continuous, uncountable space of hypotheses.
Blanchard Gilles
Roquain Etienne
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