A Quasi-residuals method in sliced inverse regression

An effective data-analytic tool, sliced inverse regression(SIR), for the analysis of multivariate data was developed by Li (Technical Report, Department of Mathematics, UCLA, 1989) and Duan and Li (J. Amer. Statist. Assoc. 86 (1991) 316). It is a method for dimension reduction. Let (Y,X) be a (p+1)-dimensional random vector, with Y[set membership, variant]R1 and X[set membership, variant]Rp. Let [Lambda] = E{Cov(XY)}. Since it is necessary for an estimate of [Lambda] in the implementation of SIR, Li (1991) considered two-sliced estimate, Hsing and Carroll (Ann. Statist. Assoc. 20 (1992) 1042) derived the asymptotic properties for the estimate and Zhu and Ng (Statist. Sinica. 5 (1995) 727) discussed fixed-number-sliced estimate. In this paper, quasi-residuals-based estimate for [Lambda] is proposed and asymptotic properties obtained. Sample properties are investigated in a simulation study.