Adaptive robust estimation in sparse vector model
Comminges, Laëtitia; Collier, Olivier; Ndaoud, Mohamed; Tsybakov, Alexandre (2021), Adaptive robust estimation in sparse vector model, Annals of Statistics, 49, 3, p. 1347-1377. 10.1214/20-AOS2002
TypeArticle accepté pour publication ou publié
Journal nameAnnals of Statistics
Institute of Mathematical Statistics
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Modélisation aléatoire de Paris X [MODAL'X]
Centre de Recherche en Économie et Statistique [CREST]
Ecole Nationale de la Statistique et de l'Analyse Economique [ENSAE]
Abstract (EN)For the sparse vector model, we consider estimation of the target vector, of its ℓ2-norm and of the noise variance. We construct adaptive estimators and establish the optimal rates of adaptive estimation when adaptation is considered with respect to the triplet “noise level—noise distribution—sparsity.” We consider classes of noise distributions with polynomially and exponentially decreasing tails as well as the case of Gaussian noise. The obtained rates turn out to be different from the minimax nonadaptive rates when the triplet is known. A crucial issue is the ignorance of the noise variance. Moreover, knowing or not knowing the noise distribution can also influence the rate. For example, the rates of estimation of the noise variance can differ depending on whether the noise is Gaussian or sub-Gaussian without a precise knowledge of the distribution. Estimation of noise variance in our setting can be viewed as an adaptive variant of robust estimation of scale in the contamination model, where instead of fixing the “nominal” distribution in advance we assume that it belongs to some class of distributions.
Subjects / Keywordsvariance estimation; functional estimation; sparsity; robust estimation; adaptivity; sub-Gaussian noise
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