Asymptotic results for sliced inverse regression

Abstract

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.