Covariate Information Matrix for Sufficient Dimension Reduction

ABSTRACT

Building upon recent research on the applications of the density information matrix, we develop a tool for sufficient dimension reduction (SDR) in regression problems called covariate information matrix (CIM). CIM exhaustively identifies the central subspace (CS) and provides a rank ordering of the reduced covariates in terms of their regression information. Compared to other popular SDR methods, CIM does not require distributional assumptions on the covariates, or estimation of the mean regression function. CIM is implemented via eigen-decomposition of a matrix estimated with a previously developed efficient nonparametric density estimation technique. We also propose a bootstrap-based diagnostic plot for estimating the dimension of the CS. Results of simulations and real data applications demonstrate superior or competitive performance of CIM compared to that of some other SDR methods. Supplementary materials for this article are available online.

KEYWORDS:

• Bootstrap
• Central subspace
• Density information matrix
• Fisher information matrix
• Nonparametric density estimation
• Sufficient dimension reduction

Acknowledgments

We gratefully acknowledge Drs. Subir Ghosh and Zhiwei Zhang of the University of California Riverside, and Drs. Bing Li, Yanyuan Ma, Matthew Reimherr, and Lingzhou Xue of Penn State, for their useful comments. We acknowledge Dr. Nikolay V. Balashov of Penn State for clarifications on the Ozone data application. We are also grateful to Drs. Yanyuan Ma (Penn State), Liping Zhu (Shanghai University), and Peng Zeng (Auburn University) for generously sharing codes, respectively, for the semiparametric, dMAVE, and Fourier sufficient dimension reduction methods. Our special thanks go to Dr. Yingcun Xia (National University of Singapore) for sharing code on an extension of the SR method still in preparation and for his insightful responses to several questions. Ge Zhao, an advisee of Dr. Ma in the Statistics graduate program at Penn State, was instrumental in adapting FORTRAN code for our simulations and real data application. Finally, we are in debt to anonymous reviewers and editors whose comments helped us greatly improve our work. During the development of this work, B. G. Lindsay passed away due to an illness. We honor the contributions of our dear friend, generous mentor, and brilliant colleague.

Additional information

Funding

W. Yao is supported by NSF grant DMS-1461677. B. G. Lindsay and D. Nandy are supported by NSF grant DMS-1407639. F. Chiaromonte is supported by NSF grant DMS-1407639.