Asymptotics For High Dimensional Regression M-Estimates: Fixed Design Results

2016/12/19 by Lei, Lihua, Bickel, Peter J., Karoui, Noureddine El · 2 citations
#FOS: Mathematics #Primary: 62J99 #Secondary: 62E20 #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1612.06358

Abstract

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 Poincaré inequality (Chatterjee, 2009) and leave-one-out analysis (El Karoui et al., 2011). 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 the ANOVA-type design, to emphasize that the technical assumptions are not just artifacts of the proof. Finally, the numerical experiments confirm and complement our theoretical results.

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