Semi-parametric efficiency bounds for high-dimensional models

2016/01/05 by Jankova, Jana, van de Geer, Sara · 1 citation
#FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1601.00815

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 semi-parametric efficiency bounds for sparse high-dimensional models, where the dimension of the parameter is larger than the sample size. We adopt a semi-parametric 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 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 ℓ₁-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.

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• High-dimensional Longitudinal Inference via a De-sparsified Dantzig-Selector

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