Asymptotic results for sliced inverse regression

It is well known that nonparametric regression techniques do not have good performance in high dimensional regression. However nonparametric regression is successful in one- or low-dimensional regression problems and is much more flexible than the parametric alternative. Hence, for high dimensional regression tasks one would like to reduce the regressor space to a lower dimension and then use nonparametric methods for curve estimation. A possible dimension reduction approach is Sliced Inverse Regression (L i 1991). It allows to find a base of a subspace in the regressor space which still carries important information for the regression. The vectors spanning this subspace are found with a technique similar to Principal Component Analysis and can be judged with the eigenvalues that belong to these vectors. Asymptotic and simulation results for the eigenvalues and vectors are presented.