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

Lihua Lei Stanford University Peter J. Bickel University of California, Berkeley Noureddine El Karoui University of California, Berkeley

Statistics Theory and Methods mathscidoc:2005.33003

Probability Theory and Related Fields, 172, 983–1079, 2018.12
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 (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.
M-estimation, robust regression, high-dimensional statistics, second order Poincar ́e inequal- ity, leave-one-out analysis.
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@inproceedings{lihua2018asymptotics,
  title={Asymptotics For High Dimensional Regression M-Estimates: Fixed Design Results},
  author={Lihua Lei, Peter J. Bickel, and Noureddine El Karoui},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200519105039111876686},
  booktitle={Probability Theory and Related Fields},
  volume={172},
  pages={983–1079},
  year={2018},
}
Lihua Lei, Peter J. Bickel, and Noureddine El Karoui. Asymptotics For High Dimensional Regression M-Estimates: Fixed Design Results. 2018. Vol. 172. In Probability Theory and Related Fields. pp.983–1079. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200519105039111876686.
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