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Minimax estimation of linear and quadratic functionals on sparsity classes

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
45
Number
3
Publisher
Institute of Mathematical Statistics
Pages
923-958
Publication identifier
10.1214/15-AOS1432
Metadata
Show full item record
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

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