The scalar-on-function modal regression for functional time series data

Abstract

This paper develops a new nonparametric estimator of the scalar-on function modal regression that is used to analyse the co-variability between a functional regressor and a scalar output variable. The new estimator inherits the smoothness of the kernel method and the robustness of the quantile regression. We assume that the functional observations are structured as a strong mixing functional time series data and we establish the almost complete consistency (with rate) of the constructed estimator. A discussion highlighting the impact of this new estimator in nonparametric functional data analysis is also given. The usefulness of this new estimator is shown using an artificial data example.

Keywords:

• Functional data analysis (FDA)
• small ball probability
• kernel method
• conditional quantile
• conditional mode
• financial time series
• functional time series
• almost complete (a.co.) convergence

Subject classifications:

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

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions which improved substantially the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We say that the sequence $\left({\mathit{\delta }}_n\right)$ converges a.co. to zero, if and only if $\mathrm{\forall }\tau >0,\sum _{n\ge 1}\mathbb{P}(|{\mathit{\delta }}_n|>\tau )<\mathrm{\infty }.$Furthermore, we say that ${\mathit{\delta }}_n=O_{a.\mathrm{co}.}\left({\mathit{\gamma }}_n\right)$, if there exists ${\tau }_0>0,$ such that $\sum _{n\ge 1}\mathbb{P}(|{\mathit{\delta }}_n|>{\tau }_0{\mathit{\gamma }}_n)<\mathrm{\infty }.$

Additional information

Funding

The authors extend also their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the Research Groups Program under grantnumber R.G.P. 1/177/44.