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Adaptive estimation in the functional nonparametric regression model

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  • Chagny, Gaëlle
  • Roche, Angelina

Abstract

In this paper, we consider nonparametric regression estimation when the predictor is a functional random variable (typically a curve) and the response is scalar. Starting from a classical collection of kernel estimates, the bias–variance decomposition of a pointwise risk is investigated to understand what can be expected at best from adaptive estimation. We propose a fully data-driven local bandwidth selection rule in the spirit of the Goldenshluger and Lepski method. The main result is a nonasymptotic risk bound which shows the optimality of our tuned estimator from the oracle point of view. Convergence rates are also derived for regression functions belonging to Hölder spaces and under various assumptions on the rate of decay of the small ball probability of the explanatory variable. A simulation study also illustrates the good practical performances of our estimator.

Suggested Citation

  • Chagny, Gaëlle & Roche, Angelina, 2016. "Adaptive estimation in the functional nonparametric regression model," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 105-118.
    • Handle: RePEc:eee:jmvana:v:146:y:2016:i:c:p:105-118
    DOI: 10.1016/j.jmva.2015.07.001
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    3. Joydeep Chowdhury & Probal Chaudhuri, 2020. "Convergence rates for kernel regression in infinite-dimensional spaces," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(2), pages 471-509, April.
    4. Kara, Lydia-Zaitri & Laksaci, Ali & Rachdi, Mustapha & Vieu, Philippe, 2017. "Data-driven kNN estimation in nonparametric functional data analysis," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 176-188.
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    6. Lydia Kara-Zaitri & Ali Laksaci & Mustapha Rachdi & Philippe Vieu, 2017. "Uniform in bandwidth consistency for various kernel estimators involving functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(1), pages 85-107, January.
    7. Boente, Graciela & Vahnovan, Alejandra, 2017. "Robust estimators in semi-functional partial linear regression models," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 59-84.
    8. Benhenni, Karim & Hassan, Ali Hajj & Su, Yingcai, 2019. "Local polynomial estimation of regression operators from functional data with correlated errors," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 80-94.
    9. Serdyukova, Nora, 2018. "Lower bounds in estimation at a point under multi-index constraint," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 183-189.

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