Sufficient Dimension Reduction (SDR) becomes an important tool for mitigating the curse of dimensionality in high dimensional regression analysis. Recently, Flexible SDR (FSDR) has been proposed to extend SDR by finding lower dimensional projections of transformed explanatory variables. The dimensions of the projections however cannot fully represent the extent of data reduction FSDR can achieve. As a consequence, optimality and other theoretical properties of FSDR are currently not well understood. In this article, we propose to use the σ-field associated with the projections, together with their dimensions to fully characterize FSDR, and refer to the σ-field as the FSDR σ-field. We further introduce the concept of minimal FSDR σ-field and consider FSDR projections with the minimal σ-field optimal. Under some mild conditions, we show that the minimal FSDR σ-field exists, attaining the lowest dimensionality at the same time. To estimate the minimal FSDR σ-field, we propose a two-stage procedure called the Generalized Kernel Dimension Reduction (GKDR) method and partially establish its consistency property under weak conditions. Extensive simulation experiments demonstrate that the GKDR method can effectively find the minimal FSDR σ-field and outperform other existing methods. The application of GKDR to a real life air pollution data set sheds new light on the connections between atmospheric conditions and air quality.
Part of Y.Z.’s work was conducted during his visits to the Center for Statistical Science, Tsinghua University, and the School of Sciences, Huzhou Normal University. L.H. acknowledges research support from the National Natural Science Foundation of China (Grant No. 12071243).
The authors would like to thank the editor, the associate editor, and the referees for their thoughtful suggestions and comments.
"Minimal σ-field for flexible sufficient dimension reduction." Electron. J. Statist. 16 (1) 1997 - 2032, 2022. https://doi.org/10.1214/22-EJS1999