Family-Wise separation rates for multiple testing

Magalie Fromont-Renoir (Université Rennes)
Wednesday, January 25, 2017 - 10:00am
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

This joint work with Matthieu Lerasle (Univ. Paris-Saclay, France) and Patricia Reynaud-Bouret (Univ. Cote d'Azur, France) is devoted to the question of the theoretical evaluation of multiple testing procedures.Where as 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 nonparametric 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 present some illustrations in various classical Gaussian
frameworks, that corroborate several expected results under particular conditions on the tested hypotheses, but also give more surprising results.