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On robust regression with high-dimensional predictors.

ABSTRACT: We study regression M-estimates in the setting where p, the number of covariates, and n, the number of observations, are both large, but p ? n. We find an exact stochastic representation for the distribution of ? = argmin(???(p)) ?(i=1)(n) ?(Y(i) - X(i')?) at fixed p and n under various assumptions on the objective function ? and our statistical model. A scalar random variable whose deterministic limit r?(?) can be studied when p/n ? ? > 0 plays a central role in this representation. We discover a nonlinear system of two deterministic equations that characterizes r?(?). Interestingly, the system shows that r?(?) depends on ? through proximal mappings of ? as well as various aspects of the statistical model underlying our study. Several surprising results emerge. In particular, we show that, when p/n is large enough, least squares becomes preferable to least absolute deviations for double-exponential errors.

SUBMITTER: El Karoui N

PROVIDER: S-EPMC3767561 | biostudies-literature | 2013 Sep

REPOSITORIES: biostudies-literature

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On robust regression with high-dimensional predictors.

El Karoui Noureddine N Bean Derek D Bickel Peter J PJ Lim Chinghway C Yu Bin B

Proceedings of the National Academy of Sciences of the United States of America 20130816 36

We study regression M-estimates in the setting where p, the number of covariates, and n, the number of observations, are both large, but p ≤ n. We find an exact stochastic representation for the distribution of β = argmin(β∈ℝ(p)) Σ(i=1)(n) ρ(Y(i) - X(i')β) at fixed p and n under various assumptions on the objective function ρ and our statistical model. A scalar random variable whose deterministic limit rρ(κ) can be studied when p/n → κ > 0 plays a central role in this representation. We discover ...[more]

PMID: 23954908

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