ADAPTIVE ESTIMATION OVER ANISOTROPIC FUNCTIONAL CLASSES VIA ORACLE APPROACH

Abstract : We address the problem of adaptive minimax estimation in white Gaus-sian noise models under L p-loss, 1 ≤ p ≤ ∞, on the anisotropic Nikol'skii classes. We present the estimation procedure based on a new data-driven selection scheme from the family of kernel estimators with varying bandwidths. For the proposed estimator we establish so-called L p-norm oracle inequality and use it for deriving minimax adaptive results. We prove the existence of rate-adaptive estimators and fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular some new asymptotics of the minimax risk are discovered, including necessary and sufficient conditions for the existence of a uniformly consistent estimator. We provide also a detailed overview of existing methods and results and formulate open problems in adaptive minimax estimation.
Type de document :
Article dans une revue
Annals of Statistics, Institute of Mathematical Statistics, 2015, 43 (3), pp.1178 - 1242. 〈10.1214/14-AOS1306〉
Liste complète des métadonnées

Littérature citée [23 références]  Voir  Masquer  Télécharger

https://hal.archives-ouvertes.fr/hal-01265235
Contributeur : Oleg Lepski <>
Soumis le : mardi 2 février 2016 - 17:26:58
Dernière modification le : lundi 4 mars 2019 - 14:04:19
Document(s) archivé(s) le : vendredi 11 novembre 2016 - 22:32:49

Fichier

lepski-est-Lp.pdf
Fichiers éditeurs autorisés sur une archive ouverte

Identifiants

Collections

Citation

Oleg Lepski. ADAPTIVE ESTIMATION OVER ANISOTROPIC FUNCTIONAL CLASSES VIA ORACLE APPROACH. Annals of Statistics, Institute of Mathematical Statistics, 2015, 43 (3), pp.1178 - 1242. 〈10.1214/14-AOS1306〉. 〈hal-01265235〉

Partager

Métriques

Consultations de la notice

162

Téléchargements de fichiers

76