February 2023 Variable selection, monotone likelihood ratio and group sparsity
Cristina Butucea, Enno Mammen, Mohamed Ndaoud, Alexandre B. Tsybakov
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Ann. Statist. 51(1): 312-333 (February 2023). DOI: 10.1214/22-AOS2251


In the pivotal variable selection problem, we derive the exact nonasymptotic minimax selector over the class of all s-sparse vectors, which is also the Bayes selector with respect to the uniform prior. While this optimal selector is, in general, not realizable in polynomial time, we show that its tractable counterpart (the scan selector) attains the minimax expected Hamming risk to within factor 2, and is also exact minimax with respect to the probability of wrong recovery. As a consequence, we establish explicit lower bounds under the monotone likelihood ratio property and we obtain a tight characterization of the minimax risk in terms of the best separable selector risk. We apply these general results to derive necessary and sufficient conditions of exact and almost full recovery in the location model with light tail distributions and in the problem of group variable selection under Gaussian noise and under more general anisotropic sub-Gaussian noise. Numerical results illustrate our theoretical findings.

Funding Statement

The work of Cristina Butucea and Alexandre Tsybakov was supported by the French National Research Agency (ANR) under the grant Labex Ecodec (ANR-11-LABEX-0047). The work of Mohamed Ndaoud is supported by a Chair of Excellence in Data Science granted by the CY Initiative.


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Cristina Butucea. Enno Mammen. Mohamed Ndaoud. Alexandre B. Tsybakov. "Variable selection, monotone likelihood ratio and group sparsity." Ann. Statist. 51 (1) 312 - 333, February 2023. https://doi.org/10.1214/22-AOS2251


Received: 1 December 2021; Revised: 1 July 2022; Published: February 2023
First available in Project Euclid: 23 March 2023

MathSciNet: MR4564858
zbMATH: 07684014
Digital Object Identifier: 10.1214/22-AOS2251

Primary: 62G07 , 62G20

Keywords: almost full recovery , exact recovery , group variable selection , Hamming loss , minimax risk , pivotal selection problem , Sparsity , Variable selection

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 1 • February 2023
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