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Biometrika Advance Access originally published online on May 23, 2007
Biometrika 2007 94(3):569-584; doi:10.1093/biomet/asm038
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Copyright © 2007 Biometrika Trust

Articles

Dimension reduction in regression without matrix inversion

R. Dennis Cook

School of Statistics, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.

Bing Li and Francesca Chiaromonte

Department of Statistics, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A.

dennis{at}stat.umn.edu

bing{at}stat.psu.edu

chiaro{at}stat.psu.edu

Received for publication 1 January 2006. Revision received 1 November 2006.
  Abstract

Regressions in which the fixed number of predictors p exceeds the number of independent observational units n occur in a variety of scientific fields. Sufficient dimension reduction provides a promising approach to such problems, by restricting attention to d < n linear combinations of the original p predictors. However, standard methods of sufficient dimension reduction require inversion of the sample predictor covariance matrix. We propose a method for estimating the central subspace that eliminates the need for such inversion and is applicable regardless of the (n, p) relationship. Simulations show that our method compares favourably with standard large sample techniques when the latter are applicable. We illustrate our method with a genomics application.

Key Words: Central subspace • {Sigma}-envelope • Singularity of sample covariance


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