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On estimation of nonsmooth functionals of sparse normal means

Collier, Olivier; Comminges, Laëtitia; Tsybakov, Alexandre B. (2020), On estimation of nonsmooth functionals of sparse normal means, Bernoulli, 26, 3, p. 1-32. 10.3150/19-BEJ1180

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Type
Article accepté pour publication ou publié
Date
2020-08
Journal name
Bernoulli
Volume
26
Number
3
Publisher
International Statistical Institute
Pages
1-32
Publication identifier
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Author(s)
Collier, Olivier
Modélisation aléatoire de Paris X [MODAL'X]
Centre de Recherche en Économie et Statistique [CREST]
Comminges, Laëtitia
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Tsybakov, Alexandre B.
Ecole Nationale de la Statistique et de l'Analyse Economique [ENSAE]
Abstract (EN)
We study the problem of estimation of the value N_gamma(theta) = sum(i=1)^d |theta_i|^gamma for 0 < gamma <= 1 based on the observations y_i = theta_i + epsilon*xi_i, i = 1,. .. , d, where theta = (theta_1,...,theta_d) are unknown parameters, epsilon > 0 is known, and xi_i are i.i.d. standard normal random variables. We establish the non-asymptotic minimax risk on the class B_0(s) of s-sparse vectors and we propose estimators achieving the minimax rate.
Subjects / Keywords
Functional estimation; nonsmooth functional; norm estimation; polynomial approximation; Sparsity