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Issue ESAIM: PS
Volume 7, 2003
Page(s) 147 - 159
DOI 10.1051/ps:2003006

ESAIM: P&S, March 2003, Vol. 7, pp. 147-159
DOI: 10.1051/ps:2003006

Adaptive tests of qualitative hypotheses

Yannick Baraud1, Sylvie Huet2 and Béatrice Laurent3

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 April 17, 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 $\mathbb{R} ^{n}$ 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.


Mathematics Subject Classification. 62G10, 62G20.

Key words: Adaptive test, test of monotonicity, test of positivity, qualitative hypothesis testing, nonparametric alternative, nonparametric regression.


© EDP Sciences, SMAI 2003


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