Adaptive tests of qualitative hypotheses

¹ École Normale Supérieure, DMA, 45 rue d'Ulm, 75230 Paris Cedex 05, France; yannick.baraud@ens.fr.
² Unité BIA, 78352 Jouy-en-Josas Cedex, France; Sylvie.Huet@jouy.inra.fr.
³ bâtiment 425, Université Paris-Sud, 91405 Orsay Cedex, France; beatrice.laurent@math.u-psud.fr.

Received: 17 April 2002

Abstract

We propose a test of a qualitative hypothesis on the mean of a n-Gaussian vector. The testing procedure is available when the variance of the observations is unknown and does not depend on any prior information on the alternative. The properties of the test are non-asymptotic. For testing positivity or monotonicity, we establish separation rates with respect to the Euclidean distance, over subsets of which are related to Hölderian balls in functional spaces. We provide a simulation study in order to evaluate the procedure when the purpose is to test monotonicity in a functional regression model and to check the robustness of the procedure to non-Gaussian errors.