HAL : hal-00722903, version 3

arXiv : 1208.1823

DOI : 10.1214/13-EJS766

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Electronic Journal of Statistics 7 (2013) 146-190

Versions disponibles :

v1 (09-08-2012)

v2 (12-08-2012)

v3 (04-01-2013)

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

Laëtitia Comminges 1, 2, Arnak Dalalyan 1, 2, 3

(24/01/2013)

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.

1 :	Laboratoire d'Informatique Gaspard-Monge (LIGM)
	Université Paris-Est Marne-la-Vallée (UPEMLV) – ESIEE – Ecole des Ponts ParisTech – Fédération de Recherche Bézout – CNRS : UMR8049
2 :	IMAGINE
	CSTB – Ecole des Ponts ParisTech – Université Paris-Est Créteil Val-de-Marne (UPEC)
3 :	Centre de Recherche en Économie et Statistique (CREST)
	INSEE – École Nationale de la Statistique et de l'Administration Économique

Domaine

:

Mathématiques/Statistiques Statistiques/Théorie

Mots Clés : variable selection – nonparametric regression – minimax rate of separation – separation constant – sharp-optimality – nonparametric testing

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Contributeur : Arnak Dalalyan <>

Soumis le : Jeudi 3 Janvier 2013, 23:20:59

Dernière modification le : Jeudi 21 Mars 2013, 22:20:36