Automatic and location-adaptive estimation in functional single-index regression

ABSTRACT

This paper develops a new automatic and location-adaptive procedure for estimating regression in a Functional Single-Index Model (FSIM). This procedure is based on k-Nearest Neighbours (kNN) ideas. The asymptotic study includes results for automatically data-driven selected number of neighbours, making the procedure directly usable in practice. The local feature of the kNN approach insures higher predictive power compared with usual kernel estimates, as illustrated in some finite sample analysis. As by-product, we state as preliminary tools some new uniform asymptotic results for kernel estimates in the FSIM model.

KEYWORDS:

• Functional data analysis
• Functional Single-Index Model
• kernel regression
• kNN regression
• uniform consistency
• semiparametric functional data analysis

AMS SUBJECT CLASSIFICATIONS:

• 62G05
• 62G08

Acknowledgments

The authors wish to thank two anonymous referees for their helpful comments and suggestions, which greatly improved the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by the Spanish Ministerio de Economía y Competitividad under Grant MTM2014-52876-R and Grant MTM2017-82724-R, in part by the Xunta de Galicia through Centro Singular de Investigación de Galicia accreditation under Grant ED431G/01 2016-2019 and through the Grupos de Referencia Competitiva under Grant ED431C2016-015, and in part by the European Union (European Regional Development Fund – ERDF). The first author also thanks the financial support from the Xunta de Galicia and the European Union (European Social Fund - ESF), the reference of which is ED481A-2018/191.