OPTIMAL ADAPTIVE ESTIMATION OF LINEAR FUNCTIONALS UNDER SPARSITY

O Collier (1) , Laëtitia Comminges (2) , A B Tsybakov (3) , Nicolas Verzelen (4)

1 MODAL'X - Modélisation aléatoire de Paris X
2 Université Paris Dauphine-PSL
3 CREST - Centre de Recherche en Économie et Statistique
4 INRA - Institut National de la Recherche Agronomique

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.

Domaines: Statistiques [math.ST]

O Collier, Laëtitia Comminges, A B Tsybakov, Nicolas Verzelen. OPTIMAL ADAPTIVE ESTIMATION OF LINEAR FUNCTIONALS UNDER SPARSITY. 2017. ⟨hal-01425801v1⟩