Dimension Reduction via Marginal Fourth Moments in Regression

Abstract

The conditional mean of the response given the predictors is often of interest in regression problems. The central mean subspace, recently introduced by Cook and Li, allows inference about aspects of the mean function in a largely nonparametric context. We propose a marginal fourth momentsmethod for estimating directions in the central mean subspace that might be missed by existing methods such as ordinary least squares (OLS) and principal Hessian directions (pHd). Our method, targeting higher order trends, particularly cubics, complements OLS and pHd because there is no inclusionamong them. Theory, estimation and inferences as well as illustrative examples are presented.