Robust estimation with modified Huber's function for functional linear models

Abstract

In this article, we consider a new robust estimation procedure for functional linear models with both slope function and functional predictor approximated by functional principal component basis functions. A modified Huber's function with tail function substituted by the exponential squared loss (ESL) is applied to the estimation procedure for achieving robustness against outliers. The tuning parameters of the new estimation method are data-driven, which enables us to reach better robustness and efficiency than other robust methods in the presence of outliers or heavy-tailed error distribution. We will show that the resulting estimator for the slope function achieves the optimal convergence rate as the least-squares estimator does in the classical functional linear regression. The convergence rate of the prediction in terms of conditional mean squared prediction error is also established. The proposed method is illustrated with simulation studies and a real data example.

Keywords:

• Robust estimation
• Huber
• exponential squared loss
• functional linear models
• functional principal component analysis

Acknowledgments

We thank the anonymous reviewers and the Associate Editor for several comments leading to a clearer presentation.

Disclosure statement

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

Notes

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Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11971001], the Beijing Natural Science Foundation [grant number 1182002], the Humanities and Social Sciences Research Projects of the Ministry of Education of China [grant number 20YJC910010],  the Fundamental Research Funds for the Universities of Henan Province [grant number NSFRF180324], the National Statistical Sciences Research Projects [grant number 2020LY075], and the Doctoral Foundation of Henan Polytechnic University [grant number B2020-37].