Annals of Statistics

Estimating sufficient reductions of the predictors in abundant high-dimensional regressions

R. Dennis Cook, Liliana Forzani, and Adam J. Rothman

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We study the asymptotic behavior of a class of methods for sufficient dimension reduction in high-dimension regressions, as the sample size and number of predictors grow in various alignments. It is demonstrated that these methods are consistent in a variety of settings, particularly in abundant regressions where most predictors contribute some information on the response, and oracle rates are possible. Simulation results are presented to support the theoretical conclusion.

Article information

Ann. Statist., Volume 40, Number 1 (2012), 353-384.

First available in Project Euclid: 4 April 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H20: Measures of association (correlation, canonical correlation, etc.)
Secondary: 62J07: Ridge regression; shrinkage estimators

Central subspace oracle property SPICE sparsity sufficient dimension reduction principal fitted components


Cook, R. Dennis; Forzani, Liliana; Rothman, Adam J. Estimating sufficient reductions of the predictors in abundant high-dimensional regressions. Ann. Statist. 40 (2012), no. 1, 353--384. doi:10.1214/11-AOS962.

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Supplemental materials

  • Supplementary material: Supplement to “Estimating sufficient reductions of the predictors in abundant high-dimensional regressions”. Owing to space constraints, we have placed the technical proofs in a supplemental article [Cook, Forzani and Rothman (2012)]. The supplement also contains several preparatory technical results that may be of interest in their own right and additional simulations. For instance, we gave in Section 7 simulation results from models with exponentially decreasing error correlations. In the supplemental article we give parallel results based on the same models but with constant error correlations.