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Smooth k NN Local Linear Estimation of the Conditional Distribution Function

Author

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  • Ibrahim M. Almanjahie

    (Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia
    Statistical Research and Studies Support Unit, King Khalid University, Abha 62529, Saudi Arabia)

  • Zouaoui Chikr Elmezouar

    (Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia
    Statistical Research and Studies Support Unit, King Khalid University, Abha 62529, Saudi Arabia)

  • Ali Laksaci

    (Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia
    Statistical Research and Studies Support Unit, King Khalid University, Abha 62529, Saudi Arabia)

  • Mustapha Rachdi

    (AGIM Team, Laboratoire AGEIS, EA 7407, Université Grenoble Alpes (France), UFR SHS, BP. 47, CEDEX 09, F38040 Grenoble, France)

Abstract

Previous works were dedicated to the functional k -Nearest Neighbors ( k NN) and the local linearity method estimations of a regression operator. In this paper, a sequence pair of ( X i , Y i ) i = 1 , … , n of functional mixing observations are considered. We treat the local linear estimation of the cumulative function of Y i given functional input variable X i . Precisely, we combine the k NN method with the local linear algorithm to construct a new and fast efficiency estimator of the conditional distribution function. The main purpose of this paper is to prove the strong convergence of the constructed estimator under mixing conditions. An application to the functional times series prediction is used to compare our proposed estimator with the existing competitive estimators, and show its efficiency and superiority.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:10:p:1102-:d:553873
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    References listed on IDEAS

    as
    1. Zouaoui Chikr-Elmezouar & Ibrahim M. Almanjahie & Ali Laksaci & Mustapha Rachdi, 2019. "FDA: strong consistency of the NN local linear estimation of the functional conditional density and mode," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 31(1), pages 175-195, January.
    2. Kudraszow, Nadia L. & Vieu, Philippe, 2013. "Uniform consistency of kNN regressors for functional variables," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1863-1870.
    3. Mustapha Rachdi & Ali Laksaci & Zoulikha Kaid & Abbassia Benchiha & Fahimah A. Al‐Awadhi, 2021. "k‐Nearest neighbors local linear regression for functional and missing data at random," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 75(1), pages 42-65, February.
    4. J. Barrientos-Marin & F. Ferraty & P. Vieu, 2010. "Locally modelled regression and functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(5), pages 617-632.
    5. Gooijer, Jan G. De & Gannoun, Ali, 2000. "Nonparametric conditional predictive regions for time series," Computational Statistics & Data Analysis, Elsevier, vol. 33(3), pages 259-275, May.
    6. Yao, Qiwei & Tong, Howell, 1995. "On initial-condition sensitivity and prediction in nonlinear stochastic systems," LSE Research Online Documents on Economics 6402, London School of Economics and Political Science, LSE Library.
    7. A. Berlinet & A. Elamine & A. Mas, 2011. "Local linear regression for functional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(5), pages 1047-1075, October.
    8. 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.
    9. 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.
    10. Ali Laksaci & Elias Ould Saïd & Mustapha Rachdi, 2021. "Uniform consistency in number of neighbors of the kNN estimator of the conditional quantile model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(6), pages 895-911, August.
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