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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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    References listed on IDEAS

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    1. Han Lin Shang, 2014. "Bayesian bandwidth estimation for a functional nonparametric regression model with mixed types of regressors and unknown error density," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(3), pages 599-615, September.
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    8. Kudraszow, Nadia L. & Vieu, Philippe, 2013. "Uniform consistency of kNN regressors for functional variables," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1863-1870.
    9. Shang, Han Lin, 2013. "Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 185-198.
    10. F. Ferraty & A. Goia & E. Salinelli & P. Vieu, 2013. "Functional projection pursuit regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 293-320, June.
    11. Antoniadis, Anestis & Paparoditis, Efstathios & Sapatinas, Theofanis, 2009. "Bandwidth selection for functional time series prediction," Statistics & Probability Letters, Elsevier, vol. 79(6), pages 733-740, March.
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    13. Amiri, Aboubacar & Crambes, Christophe & Thiam, Baba, 2014. "Recursive estimation of nonparametric regression with functional covariate," Computational Statistics & Data Analysis, Elsevier, vol. 69(C), pages 154-172.
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    15. Aneiros-Pérez, Germán & Vieu, Philippe, 2006. "Semi-functional partial linear regression," Statistics & Probability Letters, Elsevier, vol. 76(11), pages 1102-1110, June.
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    Cited by:

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    2. 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.
    3. Chaouch, Mohamed, 2019. "Volatility estimation in a nonlinear heteroscedastic functional regression model with martingale difference errors," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 129-148.
    4. 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.
    5. 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.
    6. Bouzebda, Salim & Chaouch, Mohamed, 2022. "Uniform limit theorems for a class of conditional Z-estimators when covariates are functions," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    7. 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.
    8. Omer Faruk Atac & Jeong-Eui Yun & Taehyun Noh, 2018. "Aerodynamic Design Optimization of a Micro Radial Compressor of a Turbocharger," Energies, MDPI, vol. 11(7), pages 1-18, July.
    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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