Statistica Sinica 13(2003), 297-310

AN ASYMPTOTIC THEORY FOR SIR $_{\mbox{\boldmath $\alpha$}}$ METHOD

Ali Gannoun and Jérôme Saracco

Université Montpellier II

Abstract: Sliced Inverse Regression (SIR) is a nonparametric method for achieving dimension reduction in regression problems. It is widely applicable, very easy to implement on a computer and requires no nonparametric smoothing devices such as kernel regression or smoothing splines regression. The first moment-based SIR has been extensively studied. However, one major restriction is its vulnerability to symmetric dependencies. Methods based on second moments have been suggested as a remedy, one is called SIR$_\alpha$. In this paper, we establish the asymptotic normality of the SIR$_\alpha$ estimates.

Key words and phrases: Asymptotics, eigen-elements, semiparametric regression model, sliced inverse regression (SIR).