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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 Multivariate estimation 62J05 Linear regression 15A18 Eigenvalues, singular values, and eigenvectors 65C60 Computational problems in statistics
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