Faster convergence rate for functional linear regression in reproducing kernel Hilbert spaces

ABSTRACT

Functional linear regression is in the centre of research attention involving curves as units of observations. We focus on functional linear regression in the framework of reproducing kernel Hilbert spaces studied in Cai and Yuan [Minimax and adaptive prediction for functional linear regression. J Am Stat Assoc. 2012;107(499):1201–1216]. We extend their theoretical result establishing faster convergence rate under stronger conditions which is reduced to existing results when the stronger condition is removed. In particular, our result corroborates the expectation that with smoother functions the convergence rate of the estimator is faster.

KEYWORDS:

• Convergence rate
• functional data
• reproducing kernel Hilbert space

Acknowledgments

We sincerely thank the Editor, an Associate Editor and a referee for their insightful comments that greatly improve the manuscript.

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 [No. JBK1901053, JBK1806002 and JBK140507] of China. Weiping Zhang’s research is supported by the National Natural Science Foundation of China Grant 11671374 and 71771203. Rui Li's research was supported by National Social Science fund of China [No. 17BTJ025]. The research of Heng Lian is supported by City University of Hong Kong Start-up grant 7200521, by Hong Kong RGC general research fund 11301718 and 11300519, and by Project 11871411 from NSFC and the Shenzhen Research Institute, City University of Hong Kong.