Arrêt de service programmé du vendredi 10 juin 16h jusqu’au lundi 13 juin 9h. Pour en savoir plus
Accéder directement au contenu Accéder directement à la navigation
Pré-publication, Document de travail

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

Laëtitia Comminges 1, 2 Arnak S. Dalalyan 1, 2, 3
2 imagine [Marne-la-Vallée]
CSTB - Centre Scientifique et Technique du Bâtiment, ENPC - École des Ponts ParisTech, LIGM - Laboratoire d'Informatique Gaspard-Monge
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.
Type de document :
Pré-publication, Document de travail
Liste complète des métadonnées

https://hal-enpc.archives-ouvertes.fr/hal-00722903
Contributeur : Arnak Dalalyan Connectez-vous pour contacter le contributeur
Soumis le : samedi 11 août 2012 - 00:47:13
Dernière modification le : mardi 19 octobre 2021 - 11:26:13
Archivage à long terme le : : lundi 12 novembre 2012 - 02:20:56

Fichiers

deuxiemearticle_derniere_versi...
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-00722903, version 2
  • ARXIV : 1208.1823

Collections

Citation

Laëtitia Comminges, Arnak S. Dalalyan. Minimax testing of a composite null hypothesis defined via a quadratic functional in the model of regression. 2012. ⟨hal-00722903v2⟩

Partager

Métriques

Consultations de la notice

353

Téléchargements de fichiers

342