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2019 On the asymptotic variance of the debiased Lasso
Sara van de Geer
Electron. J. Statist. 13(2): 2970-3008 (2019). DOI: 10.1214/19-EJS1599

Abstract

We consider the high-dimensional linear regression model $Y=X\beta^{0}+\epsilon$ with Gaussian noise $\epsilon$ and Gaussian random design $X$. We assume that $\Sigma:=\mathrm{I\hskip-0.48emE}X^{T}X/n$ is non-singular and write its inverse as $\Theta :=\Sigma^{-1}$. The parameter of interest is the first component $\beta_{1}^{0}$ of $\beta^{0}$. We show that in the high-dimensional case the asymptotic variance of a debiased Lasso estimator can be smaller than $\Theta_{1,1}$. For some special such cases we establish asymptotic efficiency. The conditions include $\beta^{0}$ being sparse and the first column $\Theta_{1}$ of $\Theta$ being not sparse. These sparsity conditions depend on whether $\Sigma$ is known or not.

Citation

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Sara van de Geer. "On the asymptotic variance of the debiased Lasso." Electron. J. Statist. 13 (2) 2970 - 3008, 2019. https://doi.org/10.1214/19-EJS1599

Information

Received: 1 August 2018; Published: 2019
First available in Project Euclid: 18 September 2019

zbMATH: 07113708
MathSciNet: MR4010589
Digital Object Identifier: 10.1214/19-EJS1599

Subjects:
Primary: 62J07
Secondary: 62E20

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Vol.13 • No. 2 • 2019
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