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

TL;DR

This paper introduces a new dimension reduction method in regression that combines Sliced Inverse Regression and inverse second moments, using weighted chi-squared tests to determine the subspace, outperforming existing methods.

Contribution

The paper develops a novel combined approach for dimension reduction that recovers the entire inverse second moment subspace and provides a theoretical and empirical advantage over existing methods.

Findings

Proposed method recovers the complete inverse second moment subspace.

The method shows higher power than SIR, pHd, and SAVE in simulations.

Theoretical results demonstrate the method's capability to replace multiple existing techniques.

Abstract

We propose a new method for dimension reduction in regression using the first two inverse moments. We develop corresponding weighted chi-squared tests for the dimension of the regression. The proposed method considers 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. The optimal combination may be selected based on the p-values derived from the dimension tests. Theoretically, the proposed method, 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 indicates that the proposed method may have consistently greater power than SIR, pHd, and SAVE.