A data-adaptive hybrid method for dimension reduction

Abstract

To gain the advantages of different inverse regression methods, the convex combination can be useful for estimating the central subspace. To select an appropriate combination coefficient in the hybrid method, we propose in this paper a data-adaptive hybrid method using the trace of kernel matrices. For ease of illustration, we consider particularly the combination of inverse regressions using the conditional mean and the conditional variance, both of which are separately powerful in estimating different models. Because the efficacy of the slicing estimation may deteriorate when it is used to estimate the conditional variance, we use the kernel smoother instead. The asymptotic normality at the root-n rate is achieved even with the data-driven combination weight. Illustrative examples by simulations and an application to horse mussel data is presented to demonstrate the necessity of the hybrid models and the efficacy of our kernel estimation.

Keywords:

• asymptotic normality
• central subspace
• dimension reduction
• inverse regression

AMS Subject Classification :

• 62H12
• 62J02

Acknowledgements

The authors thank the editor, the associate editor and two referees for their careful reading and constructive comments, which led to substantial improvements in the manuscript. Dr Li-Ping Zhu's work was supported by a grant from the National Social Science Foundation of China (08CTJ001), and Professor Li-Xing Zhu's research was supported by a grant from Research Grants Council of Hong Kong.