A General Theory of Nonlinear Sufficient Dimension Reduction: Formulation and Estimation

• Activity Number: 345
• Type: Contributed
• Date/Time: Tuesday, July 31, 2012 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Kuang-Yao Lee*+ and Bing Li and Francesca Chiaromonte
• Companies: Penn State University and Penn State University and Penn State University
• Keywords: Dimension reduction $\sigma$-field ; Exhaustiveness ; Generalized sliced average variance estimator ; Generalized sliced inverse regression ; Heterogeneous conditional covariance operator ; Sufficient and complete dimension reduction classes

Abstract

We give a general formulation of nonlinear sufficient dimension reduction, and explore its ramifications and scope. This formulation subsumes recent work employing reproducing kernel Hilbert spaces, and reveals many parallels between linear and nonlinear sufficient dimension reduction. We begin at the completely general level of $\sigma$-fields and this leads to the notions of sufficient, complete and sufficient, and central dimension reduction classes. We show that, when it exists, the complete and sufficient class coincides with the central class, and can be unbiasedly and exhaustively estimated by a generalized slice inverse regression estimator (GSIR). When completeness does not hold, this estimator captures only part of the central class (i.e. remains unbiased but is no longer exhaustive). However, we show that a generalized sliced average variance estimator (GSAVE) can capture a larger portion of the class. Both estimators require no numerical optimization because they can be computed by spectral decomposition of linear operators. Finally, we compare our estimators with existing methods by simulation and on actual data sets.