Sliced Inverse Moment Regression Using Weighted Chi-Squared Tests for Dimension Reduction∗

We propose a new class of dimension reduction methods using the first two inverse moments, called Sliced Inverse Moment Regression (SIMR). We develop corresponding weighted chi-squared tests for the dimension of the regression. Basically, SIMR are linear combinations of Sliced Inverse Regression (SIR) and the method using a new candidate matrix which is designed to recover the entire inverse second moment subspace. Theoretically, SIMR, as well as Sliced Average Variance Estimate (SAVE), are more capable of recovering the complete central dimension reduction subspace than SIR and Principle Hessian Directions (pHd). Therefore it can substitute for SIR, pHd, SAVE or any linear combination of them at a theoretical level. Simulation study shows that SIMR using the weighted chi-squared test may have consistently greater power than SIR, pHd, and SAVE.