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Dimension reduction in regression without matrix inversion. (English) Zbl 1135.62046

Summary: 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.

MSC:

62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
15A18 Eigenvalues, singular values, and eigenvectors
65C60 Computational problems in statistics (MSC2010)
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