WHICH BRIDGE ESTIMATOR IS THE BEST FOR VARIABLE SELECTION?

• 成果类型: Article
• 署名作者: Wang, Shuaiwen; Weng, Haolei; Maleki, Arian
• 署名单位: Columbia University; Michigan State University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1906
• 发表日期: 2020
• 页码: 2791-2823
• 关键词: information-theoretic limits; Sufficient conditions; sparsity recovery; robust regression; Lasso; asymptotics; graphs; dense; RISK

摘要

We study the problem of variable selection for linear models under the high-dimensional asymptotic setting, where the number of observations n grows at the same rate as the number of predictors p. We consider two-stage variable selection techniques (TVS) in which the first stage uses bridge estimators to obtain an estimate of the regression coefficients, and the second stage simply thresholds this estimate to select the important predictors. The asymptotic false discovery proportion (AFDP) and true positive proportion (ATPP) of these TVS are evaluated. We prove that for a fixed ATPP, in order to obtain a smaller AFDP, one should pick a bridge estimator with smaller asymptotic mean square error in the first stage of TVS. Based on such principled discovery, we present a sharp comparison of different TVS, via an in-depth investigation of the estimation properties of bridge estimators. Rather than orderwise error bounds with loose constants, our analysis focuses on precise error characterization. Various interesting signal-to-noise ratio and sparsity settings are studied. Our results offer new and thorough insights into high-dimensional variable selection. For instance, we prove that a TVS with Ridge in its first stage outperforms TVS with other bridge estimators in large noise settings; two-stage LASSO becomes inferior when the signal is rare and weak. As a by-product, we show that two-stage methods outperform some standard variable selection techniques, such as LASSO and Sure Independence Screening, under certain conditions.