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Title: Family-wise separation rates for multiple testing Authors:  Magalie Fromont - Universite Rennes 2 (France) [presenting]
Matthieu Lerasle - CNRS (France)
Patricia Reynaud-Bouret - Universite Nice Sophia-Antipolis (France)
Abstract: The question of the theoretical evaluation of multiple testing procedures is considered. Whereas many first kind error-related evaluation criteria have been defined, as generalizations or relaxations of the historical Family-Wise Error Rate (FWER), very few second kind error-related criteria have been proposed in the multiple testing literature. Starting from a parallel between some tests of multiple hypotheses and some tests of a single hypothesis, based on aggregation approaches known to lead to minimax adaptivity properties, we extend the notion of Separation Rate, at the core of the minimax theory for single hypothesis tests, to the multiple testing field. We thus introduce the notion of weak Family-Wise Separation Rate (wFWSR) and its stronger counterpart, the Family-Wise Separation Rate (FWSR), leading to an emergent minimax theory for multiple tests whose FWER is controlled. We give some illustrations in various classical Gaussian frameworks, that corroborate several expected results under particular conditions on the tested hypotheses, but that also give more surprising results.