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Nonparametric kernel regression estimation for functional stationary ergodic data: Asymptotic properties

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  • Laib, Naâmane
  • Louani, Djamal

Abstract

The aim of this paper is to study asymptotic properties of the kernel regression estimate whenever functional stationary ergodic data are considered. More precisely, in the ergodic data setting, we consider the regression of a real random variable Y over an explanatory random variable X taking values in some semi-metric abstract space. While estimating the regression function using the well-known Nadaraya-Watson estimator, we establish the consistency in probability, with a rate, as well as the asymptotic normality which induces a confidence interval for the regression function usable in practice since it does not depend on any unknown quantity. We also give the explicit form of the conditional bias term. Note that the ergodic framework is more convenient in practice since it does not need the verification of any condition as in the mixing case for example.

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  • Laib, Naâmane & Louani, Djamal, 2010. "Nonparametric kernel regression estimation for functional stationary ergodic data: Asymptotic properties," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2266-2281, November.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:10:p:2266-2281
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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.
    2. M'hamed Ezzahrioui & Elias Ould-Saïd, 2008. "Asymptotic normality of a nonparametric estimator of the conditional mode function for functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(1), pages 3-18.
    3. Th. Gasser & P. Hall & B. Presnell, 1998. "Nonparametric estimation of the mode of a distribution of random curves," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(4), pages 681-691.
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    Cited by:

    1. Mohamed Chaouch & Naâmane Laïb & Djamal Louani, 2017. "Rate of uniform consistency for a class of mode regression on functional stationary ergodic data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(1), pages 19-47, March.
    2. Ling, Nengxiang & Wang, Chao & Ling, Jin, 2016. "Modified kernel regression estimation with functional time series data," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 78-85.
    3. 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.
    4. 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.
    5. 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).
    6. Dengke Xu & Jiang Du, 2020. "Nonparametric quantile regression estimation for functional data with responses missing at random," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(8), pages 977-990, November.
    7. Mohamed Chaouch & Salah Khardani, 2015. "Randomly censored quantile regression estimation using functional stationary ergodic data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(1), pages 65-87, March.
    8. Gao, Min & Yang, Wenzhi & Wu, Shipeng & Yu, Wei, 2022. "Asymptotic normality of residual density estimator in stationary and explosive autoregressive models," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    9. Kamal Boukhetala & Jean-François Dupuy, 2019. "Modélisation Stochastique et Statistique Book of Proceedings," Post-Print hal-02593238, HAL.
    10. Laïb, Naâmane & Louani, Djamal, 2019. "Asymptotic normality of kernel density function estimator from continuous time stationary and dependent processes," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 187-196.
    11. Aneiros, Germán & Cao, Ricardo & Fraiman, Ricardo & Genest, Christian & Vieu, Philippe, 2019. "Recent advances in functional data analysis and high-dimensional statistics," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 3-9.
    12. 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.
    13. Igor S. Borisov & Yuliana Yu. Linke & Pavel S. Ruzankin, 2021. "Universal weighted kernel-type estimators for some class of regression models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(2), pages 141-166, February.
    14. Sultana Didi & Salim Bouzebda, 2022. "Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes," Mathematics, MDPI, vol. 10(22), pages 1-37, November.
    15. Didi Sultana & Louani Djamal, 2014. "Asymptotic results for the regression function estimate on continuous time stationary and ergodic data," Statistics & Risk Modeling, De Gruyter, vol. 31(2), pages 129-150, June.
    16. Akkal Fatima & Kadiri Nadia & Rabhi Abbes, 2021. "Asymptotic Normality of Conditional Density and Conditional Mode in the Functional Single Index Model," Econometrics. Advances in Applied Data Analysis, Sciendo, vol. 25(1), pages 1-24, March.
    17. Liu, Qiaojing & Zhao, Shoujiang, 2013. "Pointwise and uniform moderate deviations for nonparametric regression function estimator on functional data," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1372-1381.
    18. Chaouch, Mohamed & Laïb, Naâmane, 2019. "Optimal asymptotic MSE of kernel regression estimate for continuous time processes with missing at random response," Statistics & Probability Letters, Elsevier, vol. 154(C), pages 1-1.

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