# Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression

Abstract : We consider the problem of testing a particular type of composite null hypothesis under a nonparametric multivariate regression model. For a given quadratic functional $Q$, the null hypothesis states that the regression function $f$ satisfies the constraint $Q[f]=0$, while the alternative corresponds to the functions for which $Q[f]$ is bounded away from zero. On the one hand, we provide minimax rates of testing and the exact separation constants, along with a sharp-optimal testing procedure, for diagonal and nonnegative quadratic functionals. We consider smoothness classes of ellipsoidal form and check that our conditions are fulfilled in the particular case of ellipsoids corresponding to anisotropic Sobolev classes. In this case, we present a closed form of the minimax rate and the separation constant. On the other hand, minimax rates for quadratic functionals which are neither positive nor negative makes appear two different regimes: ''regular'' and ''irregular''. In the ''regular" case, the minimax rate is equal to $n^{-1/4}$ while in the ''irregular'' case, the rate depends on the smoothness class and is slower than in the ''regular'' case. We apply this to the issue of testing the equality of norms of two functions observed in noisy environments.
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Article dans une revue
Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2013, 7, pp.146-190. 〈10.1214/13-EJS766〉
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https://hal-enpc.archives-ouvertes.fr/hal-00722903
Contributeur : Arnak Dalalyan <>
Soumis le : jeudi 3 janvier 2013 - 23:20:59
Dernière modification le : jeudi 11 janvier 2018 - 06:20:22
Document(s) archivé(s) le : jeudi 4 avril 2013 - 03:49:20

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Laëtitia Comminges, Arnak S. Dalalyan. Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2013, 7, pp.146-190. 〈10.1214/13-EJS766〉. 〈hal-00722903v3〉

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