Asymptotics For High Dimensional Regression M-Estimates: Fixed Design Results
Abstract
We investigate the asymptotic distributions of coordinates of regression M-estimates in the moderate regime, where the number of covariates grows proportionally with the sample size . 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 Poincar\'{e} 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.
Cite
@article{arxiv.1612.06358,
title = {Asymptotics For High Dimensional Regression M-Estimates: Fixed Design Results},
author = {Lihua Lei and Peter J. Bickel and Noureddine El Karoui},
journal= {arXiv preprint arXiv:1612.06358},
year = {2016}
}