The k nearest neighbors smoothing of the relative-error regression with functional regressor

Abstract

This paper deals with the problem of the nonparametric analysis by the relative-error regression when the explanatory of a variable is of infinite dimension. Based on k-Nearest Neighbors procedure (kNN), we construct an estimator and establish its asymptotic properties. Precisely, we show its Uniform consistency in Number of Neighbors (UNN) with the precision of the convergence rate. Some empirical studies are also performed to highlight the impact of this asymptotic result in nonparametric functional statistics.

Keywords:

• Functional data analysis (FDA)
• small ball probability
• local linear method (LLM)
• spatial data
• quantile regression
• almost complete (a.co.) convergence

SUBJECT CLASSIFICATIONS:

• 62G08
• 62G10
• 62G35
• 62G07
• 62G32
• 62G30. Secondary 62H12

Acknowledgements

The authors would like to thank the Associate-Editor and the anonymous reviewer for their valuable comments and suggestions which improved substantially the quality of an earlier version of this paper.

Notes

1 A class of functions $\mathcal{C}$ is said to be a pointwise measurable class if, there exists a countable subclass ${\mathcal{C}}_0$ such that for any function $g\in \mathcal{C}$ there exists a sequence of functions ${(g_m)}_{m\in \mathsf{N}}$ in ${\mathcal{C}}_0$ such that: $|g_m(z)-g(z)|=o(1).$

2 An envelope function G for a class of functions $\mathcal{C}$ is any measurable function such that: $\underset{g\in \mathcal{C}}{\sup }|g(z)|\le G(z),$ for all z.

Additional information

Funding

The authors extend their appreciation to Deanship of Scientific Research at King Khalid University for funding this work through the Research Groups Program under grant number R.G.P. 2/67/41.