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 moments method 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 inclusion among them. Theory, estimation and inferences as well as illustrative examples are presented.
Bibliographical noteFunding Information:
The authors thank the editor, the associate editor, and two referees whose suggestions and comments led to a greatly improved article. Cook’s work was supported in part by National Science Foundation grants DMS-0103983 and DMS-0405360.
- Central mean subspaces
- Central subspace
- Dimension-reduction subspaces
- Principal Hessian directions
- Quadratic response surface models
- Regression graphics