On least favorable configurations for step-up-down tests

TL;DR

This paper examines the least favorable configurations for step-up-down multiple testing procedures, revealing that the known asymptotic properties do not extend to all intermediate procedures, and provides bounds on this phenomenon.

Contribution

It disproves the conjecture that all intermediate SUD procedures share the least favorable configuration property and extends existing calculations to two populations.

Findings

Asymptotic LFC property does not hold for all SUD procedures.

Provides explicit nonasymptotic upper bounds and rates.

Extends Steck's recursion to two populations.

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 in earlier literature that for this type of procedure, under some broad conditions, and in an asymptotical sense, the FDR is maximum when the signal strength under the alternative is maximum. In other words, so-called "Dirac uniform configurations" are asymptotically {\em least favorable} in this setting. It is known that this property also holds in a non-asymptotical sense (for any finite number of hypotheses), for the two extreme versions of SUD procedures, namely step-up and step-down (with extra conditions for the step-down case). It is therefore very natural to conjecture that this non-asymptotical {\em least favorable configuration} property could more generally be true for all "intermediate" forms of SUD procedures. We prove that this is, somewhat surprisingly, not the case. The argument is based on the exact calculations proposed earlier by Roquain and Villers (2011), that we extend here by generalizing Steck's recursion to the case of two populations. Secondly, 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.