Asymptotics for high dimensional regression M-estimates: fixed design results

• 成果类型: Article
• 署名作者: Lei, Lihua; Bickel, Peter J.; El Karoui, Noureddine
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0824-7
• 发表日期: 2018
• 页码: 983-1079
• 关键词: smallest singular-value; central-limit-theorem; robust regression; RANDOM MATRICES; EIGENVALUE; BEHAVIOR; inadmissibility; parameters; INEQUALITY; SPECTRA

摘要

We investigate the asymptotic distributions of coordinates of regression M-estimates in the moderate p/n regime, where the number of covariates p grows proportionally with the sample size n. Under appropriate regularity conditions, we establish the coordinate-wise asymptotic normality of regression M-estimates assuming a fixed-design matrix. Our proof is based on the second-order Poincare inequality and leave-one-out analysis. Some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices. We also show a counterexample, namely an ANOVA-type design, to emphasize that the technical assumptions are not just artifacts of the proof. Finally, numerical experiments confirm and complement our theoretical results.