This article is cited in 9 papers

Minimax rate of testing in sparse linear regression

A. Carpentier^a, O. Collier^bc, L. Comminges^cd, A. B. Tsybakov^e, Yu. Wang<sup>f

a</sup> University of Magdeburg, Magdeburg, Germany
^b Modal'X, Université Paris-Nanterre, Paris, France
^c CREST, Paris, France
^d CEREMADE, Université Paris-Dauphine, Paris, France
^e CREST, ENSAE, Paris, France
^f LIDS-IDSS, MIT, Cambridge, USA

Abstract: 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 $\ell_2$-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 $\ell_2$-norm of the regression parameter.

Keywords: linear regression, sparsity, signal detection.


Received: 19.07.2018
Revised: 03.10.2018
Accepted: 08.11.2018

DOI: 10.1134/S0005231019100040

 English version: Automation and Remote Control, 2019, 80:10, 1817–1834

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