On the efficiency of the de-biased Lasso

2017/08/26 by Sara van de Geer, van de Geer, Sara
Economics, Econometrics and Finance · Mathematics · #62J07 #FOS: Mathematics #Italy: Economic History and Contemporary Issues #Statistical Methods and Inference #Statistics Theory (math.ST) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1708.07986

openalex publication_date 2017/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the high-dimensional linear regression model Y = X β⁰ + ε with Gaussian noise ε and Gaussian random design X. We assume that Σ:= E X^T X / n is non-singular and write its inverse as Θ:= Σ^-1. The parameter of interest is the first component β₁⁰ of β⁰. We show that in the high-dimensional case the asymptotic variance of a debiased Lasso estimator can be smaller than Θ_1,1. For some special such cases we establish asymptotic efficiency. The conditions include β⁰ being sparse and the first column Θ₁ of Θ being not sparse. These conditions depend on whether Σ is known or not.

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