Principal single-index varying-coefficient models for dimension reduction in quantile regression

ABSTRACT

We propose a principal single-index varying-coefficient model focusing on conditional quantiles. In this general and flexible class of models, dimension reduction is achieved in three aspects: first, standard varying-coefficient models can partially avoid curse of dimensionality of large dimensional nonparametric regression; second, a one-dimensional adaptive index is constructed from multiple index variables; finally, the number of independent functions is further reduced by using principal functions. We derive the convergence rate of the estimates and asymptotic normality of the index parameter and the coefficient functions. Penalization can be added straightforwardly to obtain joint variable selection and dimension reduction. Simulations are used to demonstrate the performances and an empirical application is presented.

KEYWORDS:

• Asymptotics
• check loss
• polynomial splines
• single-index models

Acknowledgments

The authors want to thank the Editor, an associate editor and anonymous reviewers for their comments that greatly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of Fode Zhang is partially supported by the Fundamental Research Funds for the Central Universities of China [grant numbers JBK1901053, JBK1806002 and JBK140507]. Rui Li's research was supported by National Social Science fund of China [grant number 17BTJ025]. The research of Heng Lian is supported by Hong Kong RGC general research fund 11300519, and by Project 11871411 from NSFC and the Shenzhen Research Institute, City University of Hong Kong.