Dimension reduction in regression without matrix inversion

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.