Sufficient Dimension Folding for Regression Mean Function

Abstract

In this article, we consider sufficient dimension folding for the regression mean function when predictors are matrix- or array-valued. We propose a new concept named central mean dimension folding subspace and its two local estimation methods: folded outer product of gradients estimation (folded-OPG) and folded minimum average variance estimation (folded-MAVE). We establish the asymptotic properties for folded-MAVE. A modified BIC criterion is used to determine the dimensions of the central mean dimension folding subspace. We evaluate the performances of the two local estimation methods by simulated examples and demonstrate the efficacy of folded-MAVE in finite samples. And in particular, we apply our methods to analyze a longitudinal study of primary biliary cirrhosis. Supplementary materials for this article are available online.

Key Words:

• Central dimension folding subspace
• Central mean dimension folding subspace
• Minimum average variance estimation
• Modified BIC criterion

ACKNOWLEDGMENTS

We thank the Editor, the Associate Editor, and two referees whose suggestions led to a greatly improved article. Y. Xue's work was supported in part by NSF-China grant 11401095 and Program for Innovative Research Team at UIBE. X. Yin’s work was supported in part by an NSF grant 1205546. This study was also supported in part by the resource of The Georgia Advanced Computing Research Center, collaboration between the Office of the Vice President for Research and the Office of the Chief Information Officer.