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

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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  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.

Keywords

M-estimation Robust regression High-dimensional statistics Second order Poincaré inequality Leave-one-out analysis 

Mathematics Subject Classification

Primary 62J99 Secondary 62E20 

Supplementary material

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of California, BerkeleyBerkeleyUSA
  2. 2.Criteo ResearchParisFrance

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