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Optimal adaptive estimation of linear functionals under sparsity

Collier, Olivier; Comminges, Laëtitia; Tsybakov, Alexandre; Verzelen, Nicolas (2018), Optimal adaptive estimation of linear functionals under sparsity, Annals of Statistics, 46, 6, p. 3130 - 3150. 10.1214/17-AOS1653

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Type
Article accepté pour publication ou publié
Date
2018
Journal name
Annals of Statistics
Volume
46
Number
6
Publisher
Institute of Mathematical Statistics
Pages
3130 - 3150
Publication identifier
Metadata
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Author(s)
Collier, Olivier
Modélisation aléatoire de Paris X [MODAL'X]
Comminges, Laëtitia
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Tsybakov, Alexandre
Centre de Recherche en Économie et Statistique [CREST]
Verzelen, Nicolas
Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie [MISTEA]
Abstract (EN)
We consider the problem of estimation of a linear functional in the Gaussian sequence model where the unknown vector theta is an element of R-d belongs to a class of s-sparse vectors with unknown s. We suggest an adaptive estimator achieving a nonasymptotic rate of convergence that differs from the minimax rate at most by a logarithmic factor. We also show that this optimal adaptive rate cannot be improved when s is unknown. Furthermore, we address the issue of simultaneous adaptation to s and to the variance sigma(2) of the noise. We suggest an estimator that achieves the optimal adaptive rate when both s and sigma(2) are unknown.
Subjects / Keywords
Nonasymptotic minimax estimation; adaptive estimation; linear functional; sparsity; unknown noise variance