Necessary and sufficient conditions for variable selection consistency of the Lasso in high dimensions
type: Item
instance of: scholarly article
title: Necessary and sufficient conditions for variable selection consistency of the Lasso in high dimensions
zbMATH Open document ID: 1469.62307
DOI: 10.1214/20-AOS1979
published in: The Annals of Statistics
publication date: 2021-07-05T00:00:00Z
review text: Consider the multivariate linear regression model \[ y_i=\mathbf{X}^\prime_i\boldsymbol{\beta}+\epsilon_i\,, \] for \(i=1,\ldots,n\), where \(\mathbf{X}_i\) and \(\boldsymbol{\beta}\) are \(p\)-dimensional vectors containing the covariates and regression parameters, respectively, and the \(\epsilon_i\) are IID errors with mean zero. Assume that only \(p_0\) of the \(p\) parameters are nonzero, where both \(p\) and \(p_0\) may grow with the sample size \(n\). In the context of the Lasso method of penalized regression, and under some regularity and boundedness conditions, the author provides necessary and sufficient conditions for Lasso to be variable selection consistent (i.e., to choose correctly which parameters are nonzero with probability tending to 1 as \(n\to\infty\)), which show that this property continues to hold for a wide range of penalty terms and \(p\) potentially very large compared to \(n\). They also show that in many cases the Lasso estimators cannot be both variable selection consistent and \(\sqrt{n}\)-consistent.
reviewed by: Fraser Daly
Mathematics Subject Classification ID: 62J07
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62J05
zbMATH DE Number: 7367863
zbMATH Keywords: asymptotic normality
zbMATH Keywords: irrepresentable condition
zbMATH Keywords: oracle property
zbMATH Keywords: regularization
author: Soumendra N. Lahiri
describes a project that uses: glmnet
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7367863