Optimal adaptive estimation of linear functionals under sparsity

Olivier Collier (1) , Laëtitia Comminges (2, 3) , Alexandre B. Tsybakov (4) , Nicolas Verzelen (5)

1 MODAL'X - Modélisation aléatoire de Paris X
2 Université Paris Dauphine-PSL
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 CREST - Centre de Recherche en Économie et Statistique
5 MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie

We consider the problem of estimation of a linear functional in the Gaussian sequence model where the unknown vector θ ∈ R^d belongs to a class of s-sparse vectors with unknown s. We suggest an adaptive estimator achieving a non-asymptotic 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 σ^2 of the noise. We suggest an estimator that achieves the optimal adaptive rate when both s and σ^2 are unknown.

Mots clés: Nonasymptotic minimax estimation; adaptive estimation; linear functional; sparsity; unknown noise variance

Domaines: Statistiques [math.ST]; Optimisation et contrôle [math.OC]

Olivier Collier, Laëtitia Comminges, Alexandre B. Tsybakov, Nicolas Verzelen. Optimal adaptive estimation of linear functionals under sparsity. Annals of Statistics, 2018, 46 (6A), pp.3130-3150. ⟨10.1214/17-AOS1653⟩. ⟨hal-01425801v2⟩