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Asymptotic normality of locally modelled regression estimator for functional data

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  • Zhiyong Zhou
  • Zhengyan Lin
Abstract
We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As an alternative to the well-known Nadaraya-Watson estimator for regression function in this framework, the locally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu, P. (2010), ‘Locally Modelled Regression and Functional Data’, Journal of Nonparametric Statistics , 22, 617--632]. In this paper, the asymptotic properties of locally modelled regression estimator for functional data are considered. The mean-squared convergence as well as asymptotic normality for the estimator are established. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function and derive the Wilk's phenomenon for the empirical likelihood inference. Furthermore, a simulation study is presented to illustrate our theoretical results.

Suggested Citation

  • Zhiyong Zhou & Zhengyan Lin, 2016. "Asymptotic normality of locally modelled regression estimator for functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(1), pages 116-131, March.
    • Handle: RePEc:taf:gnstxx:v:28:y:2016:i:1:p:116-131
    DOI: 10.1080/10485252.2015.1114112
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    References listed on IDEAS

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    1. Masry, Elias, 2005. "Nonparametric regression estimation for dependent functional data: asymptotic normality," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 155-177, January.
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    6. Benhenni, K. & Hedli-Griche, S. & Rachdi, M. & Vieu, P., 2008. "Consistency of the regression estimator with functional data under long memory conditions," Statistics & Probability Letters, Elsevier, vol. 78(8), pages 1043-1049, June.
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    Cited by:

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    2. Somia Ayad & Ali Laksaci & Saâdia Rahmani & Rachida Rouane, 2020. "On the local linear modelization of the conditional density for functional and ergodic data," METRON, Springer;Sapienza Università di Roma, vol. 78(2), pages 237-254, August.
    3. Ibrahim M. Almanjahie & Zouaoui Chikr Elmezouar & Ali Laksaci & Mustapha Rachdi, 2021. "Smooth k NN Local Linear Estimation of the Conditional Distribution Function," Mathematics, MDPI, vol. 9(10), pages 1-14, May.
    4. Oussama Bouanani & Saâdia Rahmani & Ali Laksaci & Mustapha Rachdi, 2020. "Asymptotic normality of conditional mode estimation for functional dependent data," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(2), pages 465-481, June.
    5. Saâdia Rahmani & Oussama Bouanani, 2023. "Local linear estimation of the conditional cumulative distribution function: Censored functional data case," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 741-769, February.

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