Minimax rate of testing in sparse linear regression

Alexandra Carpentier (1) , Olivier Collier (2) , Laëtitia Comminges (3) , Alexandre Tsybakov (4, 5) , Yuhao Wang (6)

1 Statistical Laboratory [Cambridge]
2 MODAL'X - Modélisation aléatoire de Paris X
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 CREST - Centre de Recherche en Economie et Statistique [Bruz]
5 ENSAE - Ecole Nationale de la Statistique et de l'Analyse Economique
6 Nanyang Technological University [Singapour]

We consider the problem of testing the hypothesis that the parameter of linear regression model is 0 against an s-sparse alternative separated from 0 in the 2-distance. We show that, in Gaussian linear regression model with p < n, where p is the dimension of the parameter and n is the sample size, the non-asymptotic minimax rate of testing has the form sqrt((s/n) log(1 + sqrt(p)/s)). We also show that this is the minimax rate of estimation of the 2-norm of the regression parameter. MSC 2010 subject classifications: 62J05, 62G10.

Mots clés: sparsity; signal detection; linear regression; Gaussian sequence model

Domaines: Statistiques [stat]

Alexandra Carpentier, Olivier Collier, Laëtitia Comminges, Alexandre Tsybakov, Yuhao Wang. Minimax rate of testing in sparse linear regression. Automation and Remote Control / Avtomatika i Telemekhanika, 2019. ⟨hal-01770434⟩