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October 2018 Semiparametric efficiency bounds for high-dimensional models
Jana Janková, Sara van de Geer
Ann. Statist. 46(5): 2336-2359 (October 2018). DOI: 10.1214/17-AOS1622

Abstract

Asymptotic lower bounds for estimation play a fundamental role in assessing the quality of statistical procedures. In this paper, we propose a framework for obtaining semiparametric efficiency bounds for sparse high-dimensional models, where the dimension of the parameter is larger than the sample size. We adopt a semiparametric point of view: we concentrate on one-dimensional functions of a high-dimensional parameter. We follow two different approaches to reach the lower bounds: asymptotic Cramér–Rao bounds and Le Cam’s type of analysis. Both of these approaches allow us to define a class of asymptotically unbiased or “regular” estimators for which a lower bound is derived. Consequently, we show that certain estimators obtained by de-sparsifying (or de-biasing) an $\ell_{1}$-penalized M-estimator are asymptotically unbiased and achieve the lower bound on the variance: thus in this sense they are asymptotically efficient. The paper discusses in detail the linear regression model and the Gaussian graphical model.

Citation

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Jana Janková. Sara van de Geer. "Semiparametric efficiency bounds for high-dimensional models." Ann. Statist. 46 (5) 2336 - 2359, October 2018. https://doi.org/10.1214/17-AOS1622

Information

Received: 1 June 2016; Revised: 1 August 2017; Published: October 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06964335
MathSciNet: MR3845020
Digital Object Identifier: 10.1214/17-AOS1622

Subjects:
Primary: 62J07
Secondary: 62F12

Keywords: Asymptotic efficiency , Cramér–Rao bound , graphical models , high-dimensional , Lasso , Le Cam’s lemma , Linear regression , Sparsity

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 5 • October 2018
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