An asymptotic theory for Sliced Inverse Regression

Sliced Inverse Regression (S.I.R.) is a method for reducing the dimension of the explanatory variable x in nonparametric regression problems. Li (1991) considers a general regression model of the form y = g(x'beta(1)...,x'beta(K),epsilon) with an arbitrary and unknown link function g, and studies a link-free and distribution-free method for estimating E, the space spanned by the beta(k)`s, called the effective dimension reduction (e.d.r.) space. It is widely applicable, easy to implement on a computer and requires no nonparametric smoothing devices such as kernel regression. The method begins with a partition of the range of y into a fixed number of slices. Let us denote T(.) this partition. The conditional mean of x given T(y) is then estimated by the sample mean within each slice. After that the covariance matrix of the conditional mean, Gamma(T) := var(E[[x\T(y)]), is estimated by <(Gamma)over cap>(T), the sample covariance matrix of all slice's conditional means. Let us denote Sigma (resp. <(Sigma)over cap>) the theoretical (resp. sample) covariance matrix of x. Finally the It eigenvectors associated with the largest K eigenvalues of Sigma(-1)Gamma(T) span the e.d.r, space, and the K eigenvectors associated with the largest K eigenvalues of <(Sigma)over cap>(-1)<(Gamma)over cap>(T) give an estimate of a basis of E. In this paper, we establish the asymptotic distribution of the aforedefined estimator of a basis of E. The asymptotic distributions of the associated eigenprojector, eigenvalues and eigenvectors are obtained.