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On the Validity of the Bootstrap in Non‐Parametric Functional Regression

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  • FRÉDÉRIC FERRATY
  • INGRID VAN KEILEGOM
  • PHILIPPE VIEU

Abstract

. We consider the functional non‐parametric regression model Y= r(χ)+ɛ, where the response Y is univariate, χ is a functional covariate (i.e. valued in some infinite‐dimensional space), and the error ɛ satisfies E(ɛ | χ) = 0. For this model, the pointwise asymptotic normality of a kernel estimator of r (·) has been proved in the literature. To use this result for building pointwise confidence intervals for r (·), the asymptotic variance and bias of need to be estimated. However, the functional covariate setting makes this task very hard. To circumvent the estimation of these quantities, we propose to use a bootstrap procedure to approximate the distribution of . Both a naive and a wild bootstrap procedure are studied, and their asymptotic validity is proved. The obtained consistency results are discussed from a practical point of view via a simulation study. Finally, the wild bootstrap procedure is applied to a food industry quality problem to compute pointwise confidence intervals.

Suggested Citation

  • Frédéric Ferraty & Ingrid Van Keilegom & Philippe Vieu, 2010. "On the Validity of the Bootstrap in Non‐Parametric Functional Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(2), pages 286-306, June.
    • Handle: RePEc:bla:scjsta:v:37:y:2010:i:2:p:286-306
    DOI: 10.1111/j.1467-9469.2009.00662.x
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    References listed on IDEAS

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    Cited by:

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    3. Martínez-Camblor, Pablo & Corral, Norberto, 2011. "Repeated measures analysis for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 55(12), pages 3244-3256, December.
    4. Geenens, Gery, 2015. "Moments, errors, asymptotic normality and large deviation principle in nonparametric functional regression," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 369-377.
    5. González-Rodríguez, Gil & Colubi, Ana, 2017. "On the consistency of bootstrap methods in separable Hilbert spaces," Econometrics and Statistics, Elsevier, vol. 1(C), pages 118-127.
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    8. Liebl, Dominik, 2019. "Inference for sparse and dense functional data with covariate adjustments," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 315-335.
    9. Han Shang, 2014. "Bayesian bandwidth estimation for a semi-functional partial linear regression model with unknown error density," Computational Statistics, Springer, vol. 29(3), pages 829-848, June.
    10. Ferraty, F. & Van Keilegom, I. & Vieu, P., 2012. "Regression when both response and predictor are functions," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 10-28.
    11. Shang, Han Lin, 2016. "A Bayesian approach for determining the optimal semi-metric and bandwidth in scalar-on-function quantile regression with unknown error density and dependent functional data," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 95-104.
    12. Yousri Slaoui, 2021. "Recursive non-parametric kernel classification rule estimation for independent functional data," Computational Statistics, Springer, vol. 36(1), pages 79-112, March.
    13. Graciela Boente & Daniela Rodriguez & Mariela Sued, 2018. "Testing equality between several populations covariance operators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 919-950, August.
    14. Pini, Alessia & Stamm, Aymeric & Vantini, Simone, 2018. "Hotelling’s T2 in separable Hilbert spaces," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 284-305.
    15. 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.
    16. Krebs, Johannes T.N., 2018. "A large deviation inequality for β-mixing time series and its applications to the functional kernel regression model," Statistics & Probability Letters, Elsevier, vol. 133(C), pages 50-58.
    17. Krebs, Johannes T.N., 2019. "The bootstrap in kernel regression for stationary ergodic data when both response and predictor are functions," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 620-639.
    18. Laurent Delsol, 2013. "No effect tests in regression on functional variable and some applications to spectrometric studies," Computational Statistics, Springer, vol. 28(4), pages 1775-1811, August.
    19. Zhiqiang Jiang & Zhensheng Huang & Jing Zhang, 2023. "Functional single-index composite quantile regression," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(5), pages 595-603, July.
    20. Yousri Slaoui, 2020. "Recursive nonparametric regression estimation for dependent strong mixing functional data," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 665-697, October.
    21. Germán Aneiros & Paula Raña & Philippe Vieu & Juan Vilar, 2018. "Bootstrap in semi-functional partial linear regression under dependence," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(3), pages 659-679, September.
    22. Ling, Nengxiang & Xu, Qian, 2012. "Asymptotic normality of conditional density estimation in the single index model for functional time series data," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2235-2243.

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