SEMIPARAMETRIC EFFICIENCY BOUNDS FOR HIGH-DIMENSIONAL MODELS

• 成果类型: Article
• 署名作者: Jankova, Jana; van de Geer, Sara
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1622
• 发表日期: 2018
• 页码: 2336-2359
• 关键词: confidence-intervals; Lasso; regression; selection

摘要

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 Cramer-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 l(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.