Statistica Sinica 24 (2014), 1-23

ON LEAST FAVORABLE CONFIGURATIONS

FOR STEP-UP-DOWN TESTS

Gilles Blanchard$^1$, Thorsten Dickhaus$^2$, Étienne Roquain$^3$ and Fanny Villers$^3$

$^1$Potsdam University, $^2$Humboldt University and $^3$UPMC University

Abstract: This paper investigates an open issue related to false discovery rate (FDR) control of step-up-down (SUD) multiple testing procedures. It has been established that for this type of procedure, under some broad conditions and in an asymptotic sense, the FDR is maximum when the signal strength under the alternative is maximum. In other words, so-called ``Dirac uniform configurations" are asymptotically least favorable in this setting. It is known that this property also holds in a nonasymptotic sense (for any finite number of hypotheses) for the two extreme versions of SUD procedures, namely step-up and step-down (under additional conditions for the step-down case). It is therefore natural to conjecture that this nonasymptotic least favorable configuration property could more generally be true for ``intermediate'' forms of SUD procedures. We prove that this is not the case. The argument is based on the exact calculations proposed earlier by Roquain and Villers (2011a); we extend them by generalizing Steck's recursion to the case of two populations. Furthermore, we quantify the magnitude of this phenomenon by providing a nonasymptotic upper bound and explicit vanishing rates as a function of the total number of hypotheses.

Key words and phrases: False discovery rate, least favorable configuration, multiple testing, Steck's recursions, step-up-down.