Minimax estimation of linear and quadratic functionals on sparsity classes

Collier, Olivier; Comminges, Laëtitia; Tsybakov, Alexandre

Collier, Olivier; Comminges, Laëtitia; Tsybakov, Alexandre (2017), Minimax estimation of linear and quadratic functionals on sparsity classes, Annals of Statistics, 45, 3, p. 923-958. 10.1214/15-AOS1432

Type

Article accepté pour publication ou publié

Date

2017

Journal name

Annals of Statistics

Volume

Number

Publisher

Institute of Mathematical Statistics

Pages

923-958

Publication identifier

10.1214/15-AOS1432

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
Ecole Nationale de la Statistique et de l'Analyse Economique [ENSAE]

Abstract (EN)

For the Gaussian sequence model, we obtain nonasymptotic minimax rates of estimation of the linear, quadratic and the ℓ2-norm functionals on classes of sparse vectors and construct optimal estimators that attain these rates. The main object of interest is the class B0(s) of s-sparse vectors θ=(θ1,…,θd), for which we also provide completely adaptive estimators (independent of s and of the noise variance σ) having logarithmically slower rates than the minimax ones. Furthermore, we obtain the minimax rates on the ℓq-balls Bq(r)={θ∈Rd:∥θ∥q≤r} where 0

Subjects / Keywords

linear functional; Nonasymptotic minimax estimation; quadratic functional; Sparsity; thresholding; unknown noise variance