The k nearest neighbors local linear estimator of semi functional partial linear model with missing response at random

Amina Naceri; Tawfik Benchikh; Ibrahim M. Almanjahie; Omar Fetitah; Mohammed Kadi Attouch; Fatimah Alshahrani; Amina Naceri; Tawfik Benchikh; Ibrahim M. Almanjahie; Omar Fetitah; Mohammed Kadi Attouch; Fatimah Alshahrani

doi:10.3934/math.2025714

AIMS Mathematics

2025, Volume 10, Issue 7: 15929-15954. doi: 10.3934/math.2025714

Previous Article Next Article

Research article

The k nearest neighbors local linear estimator of semi functional partial linear model with missing response at random

1.
Laboratory of Statistics and Stochastic Processes, University of Djillali Liabes BP 89, Sidi Bel Abbes 22000, Algeria; Email: benchikh.tawfik@gmail.com, attou_kadi@yahoo.fr
2.
Medical Faculty, Djillali Liabes University BP 89, Sidi Bel Abbes, 22000, Algeria
3.
Ecole Supérieure en Informatique, Sidi Bel Abbes, 22000, Algeria; Email: am.naceri@esi-sba.dz, o.fetitah@esi-sba.dz
4.
Department of Mathematics, College of Science, King Khalid University, Abha 62223, Saudi Arabia; Email: imalmanjahi@kku.edu.sa
5.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia; Email: fmalshahrani@pnu.edu.sa

Received: 24 March 2025 Revised: 17 June 2025 Accepted: 08 July 2025 Published: 15 July 2025
MSC : 60G25, 62G05, 62G08, 62G20

This paper aims to investigate a semi-functional partial linear regression model in the presence of missing data in the response variable under the missing at random mechanism. We construct estimators using the kNN-local linear method and establish the asymptotic distribution of the parametric component. Additionally, the uniform almost complete consistency rates for the nonparametric component with respect to the number of neighbors under appropriate conditions is derived. Through simulations and real data analysis, we assess the effectiveness of the proposed approach and demonstrate its superiority by comparing it with existing methods for semi-functional partial linear regression models.
- functional data analysis,
- partial linear regression,
- missing at random data,
- kNN estimation,
- local linear estimation
Citation: Amina Naceri, Tawfik Benchikh, Ibrahim M. Almanjahie, Omar Fetitah, Mohammed Kadi Attouch, Fatimah Alshahrani. The k nearest neighbors local linear estimator of semi functional partial linear model with missing response at random[J]. AIMS Mathematics, 2025, 10(7): 15929-15954. doi: 10.3934/math.2025714

Related Papers:

Abstract

This paper aims to investigate a semi-functional partial linear regression model in the presence of missing data in the response variable under the missing at random mechanism. We construct estimators using the kNN-local linear method and establish the asymptotic distribution of the parametric component. Additionally, the uniform almost complete consistency rates for the nonparametric component with respect to the number of neighbors under appropriate conditions is derived. Through simulations and real data analysis, we assess the effectiveness of the proposed approach and demonstrate its superiority by comparing it with existing methods for semi-functional partial linear regression models.

References

[1]	J. Ramsay, B. Silverman, Functional data analysis, 2 Eds., Spinger-Verlag, New York, 2005.
[2]	P. Kokoszka, M. Reimherr, Introduction to functional data analysis, 1st Edition, eBook, Chapman and Hall, 2017. https://doi.org/10.1201/9781315117416
[3]	T. Hsing, R. L. Eubank, Theoretical foundations of functional data analysis, with an introduction to linear operators, John Wiley and Sons, 2015.
[4]	G. Aneiros-Pérez, R. Cao, R. Fraiman, C. Genest, P. Vieu, Recent advances in functional data analysis and high-dimensional statistics, J. Multivariate Anal., 170 (2019), 3–9. https://doi.org/10.1016/j.jmva.2018.11.007 doi: 10.1016/j.jmva.2018.11.007
[5]	G. Aneiros-Pérez, I. Horov$\acute{a}$, M. Hu$\tilde{s}$kov$\acute{a}$, P. Vieu, Editorial for the special issue on functional data analysis and related fields, J. Multivariate Anal., 189 (2022). https://doi.org/10.1016/j.jmva.2021.104861
[6]	M. Rachdi, Functional data analysis: Theory and applications to different scenarios, Mathematics, an Open Access Journal by MDPI 2023. Avaiable from: https://www.mdpi.com/journal/mathematics/specialissues/45POZ9BG9S.
[7]	F. Ferraty, F. Vieu, Nonparametric functional data analysis. Theory and Practice, Springer Series in Statistics, New York, 2006.
[8]	A. Cuevas, A partial overview of the theory of statistics with functional data, J. Statist. Plann. Inference, 147 (2014), 1–23. https://doi.org/10.1016/j.jspi.2013.04.002 doi: 10.1016/j.jspi.2013.04.002
[9]	S. Greven, F. Scheipl, A general framework for functional regression modelling, Stat. Model., 17 (2017), 1–35. https://doi.org/10.1177/1471082X16681317 doi: 10.1177/1471082X16681317
[10]	G. Aneiros-Pérez, P. Vieu, Semi-functional partial linear regression, Stat. Probab. Lett., 76 (2006), 1102–1110. https://doi.org/10.1016/j.spl.2005.12.007 doi: 10.1016/j.spl.2005.12.007
[11]	G. Boente, A. Vahnovan, Robust estimators in semi-functional partial linear regression models, J. Multivariate Anal., 154 (2017), 59–84. https://doi.org/10.1016/j.jmva.2016.10.005 doi: 10.1016/j.jmva.2016.10.005
[12]	S. Feng, L. Xue, Partially functional linear varying coefficient model, Statistics, 50 (2016), 717–732. https://doi.org/10.1080/02331888.2016.1138954 doi: 10.1080/02331888.2016.1138954
[13]	N. Ling, G. Aneiros-Pérez, P. Vieu, kNN estimation in functional partial linear modeling, Statist. Papers, 61 (2020), 423–444. https://doi.org/10.1007/s00362-017-0946-0 doi: 10.1007/s00362-017-0946-0
[14]	G. Aneiros-Pérez, P. Vieu, Nonparametric time series prediction: A semi-functional partial linear modeling, J. Multivariate Anal., 99 (2008), 834–857. https://doi.org/10.1016/j.jmva.2007.04.010 doi: 10.1016/j.jmva.2007.04.010
[15]	M. Benallou, M. K. Attouch, T. Benchikh, O. Fetitah, Asymptotic results of semi-functional partial linear regression estimate under functional spatial dependency, Commun. Stat.-Theor. Method., 51 (2021), 1–21. https://doi.org/10.1080/03610926.2020.1871021 doi: 10.1080/03610926.2020.1871021
[16]	N. Ling, P. Vieu, Nonparametric modelling for functional data: Selected survey and tracks for future, Statistics, 52 (2018), 934–949. https://doi.org/10.1080/02331888.2018.1487120 doi: 10.1080/02331888.2018.1487120
[17]	N. Ling, P. Vieu, On semiparametric regression in functional data analysis, WIRES Comput. Stat., 12 (2020), 20–30. https://doi.org/10.1002/wics.1538 doi: 10.1002/wics.1538
[18]	S. Novo, G. Aneiros-Pérez, P. Vieu, A kNN procedure in semiparametric functional data analysis, Statist. Probab. Lett., 171 (2021). https://doi.org/10.1016/j.spl.2020.109028
[19]	N. H. Kedir, T. Benchikh, A. Naceri, O. Fetitah, Local linear-kNN smoothing for semi-functional partial linear regression, Hacet. J. Math. Stat., 53 (2024), 537–555. https://doi.org/10.15672/hujms.1294382 doi: 10.15672/hujms.1294382
[20]	M. K. Attouch, A. Laksaci, F. Rafaa, Estimation locale linéaire de la régression non paramétrique fonctionnelle par la méthode des k plus proches voisins, CR Math., 355 (2017), 824–829. https://doi.org/10.1016/j.crma.2017.05.007 doi: 10.1016/j.crma.2017.05.007
[21]	I. M. Almanjahie, Z. Chikr-Elmezouar, A. Laksaci, M. Rachdi, kNN local linear estimation of the conditional cumulative distribution function: Dependent functional data case, C. R. Acad. Sci. Paris, Ser. I, 356 (2018), 1036–1039. https://doi.org/10.1016/j.crma.2018.09.001 doi: 10.1016/j.crma.2018.09.001
[22]	Z. Chikr-Elmezouar, I. M. Almanjahie, A. Laksaci, M. Rachdi, FDA: Strong consistency of the kNN local linear estimation of the functional conditional density and mode, J. Nonparametr. Stat., 31 (2019), 175–195. https://doi.org/10.1080/10485252.2018.1538450 doi: 10.1080/10485252.2018.1538450
[23]	I. M. Almanjahie, W. Mesfer, A. Laksaci, M. Rachdi, Computational aspects of the kNN local linear smoothing for some conditional models in high dimensional statistics, Commun. Stat. Simul. Comput., 52 (2023), 2985–3005. https://doi.org/10.1080/03610918.2021.1923745 doi: 10.1080/03610918.2021.1923745
[24]	F. Alshahrani, W. Bouabsa, I. M. Almanjahie, M. K. Attouch, kNN local linear estimation of the conditional density and mode for functional spatial high dimensional data, AIMS Math., 8 (2023), 15844–15875. https://doi.org/10.3934/math.2023809 doi: 10.3934/math.2023809
[25]	F. Ferraty, F. Sued, P. Vieu, Mean estimation with data missing at random for functional covariables, Statistics, 47 (2013), 688–706. https://doi.org/10.1080/02331888.2011.650172 doi: 10.1080/02331888.2011.650172
[26]	N. Ling, L. Liang, P. Vieu, Nonparametric regression estimation for functional stationary ergodic data with missing at random, J. Stat. Plan. Inference, 162 (2015), 75–87. https://doi.org/10.1016/j.jspi.2015.02.001 doi: 10.1016/j.jspi.2015.02.001
[27]	N. Ling, Y. Liu, P. Vieu, Conditional mode estimation for functional stationary ergodic data with responses missing at random, Statistics, 50 (2016), 991–1013. https://doi.org/10.1080/02331888.2015.1122012 doi: 10.1080/02331888.2015.1122012
[28]	F. Alshahrani, I. M. Almanjahi, T. Benchikh, O. Fetitah, M. K. Attouch, Asymptotic normality of nonparametric kernel regression estimation for missing at random functional spatial data, J. Math., 2 (2023), 1–20. https://doi.org/10.1155/2023/8874880.4/math.2023809 doi: 10.1155/2023/8874880.4/math.2023809
[29]	M. Rachdi, A. Laksaci, K. Kaid, A. Benchiha, F. Al-Awadh, k-Nearest neighbors local linear regression for functional and missing data at random, Stat. Neerl., 75 (2021), 42–65. https://doi.org/10.1111/stan.12224 doi: 10.1111/stan.12224
[30]	I. M. Almanjahie, W. M. Alahmari, A. Laksaci, The k nearest neighbors local linear estimator of functional conditional density when there are missing data, Hacet. J. Math. Stat., 51 (2022), 914–931. https://doi.org/10.15672/hujms.796694 doi: 10.15672/hujms.796694
[31]	N. Ling, R. Kan, P. Vieu, S. Meng, Semi-functional partially linear regression model with responses missing at random, Metrika, 82 (2019), 39–70. https://doi.org/10.1007/s00184-018-0688-6 doi: 10.1007/s00184-018-0688-6
[32]	T. Benchikh, I. M. Almanjahie, O. Fetitah, M. K. Attouch, Estimation for spatial semi-functional partial linear regression model with missing response at random, Demonstr. Math., 58 (2025), https://doi.org/10.1515/dema-2025-0108
[33]	A. Naceri, A. Laksaci, M. Rachdi, Exact quadratic error of the local linear regression operator estimator for functional covariates, In: Functional statistics and applications, Springer Cham Heidelberg, New York, 2015, 79–90. https://doi.org/10.1007/978-3-319-22476-3_5
[34]	N. Kudraszow, P. Vieu, Uniform consistency of kNN regressors for functional variables, Statist. Probab. Lett., 83 (2013), 1863–1870. https://doi.org/10.1016/j.spl.2013.04.017 doi: 10.1016/j.spl.2013.04.017
[35]	F. Ferraty, A. Mas, P. Vieu, Nonparametric regression on functional data: Inference and practical aspects, Aust. N. Z. J. Stat., 49 (2007), 267–286. https://doi.org/10.1111/j.1467-842X.2007.00480.x doi: 10.1111/j.1467-842X.2007.00480.x
[36]	J. Barrientos-Marin, F. Ferraty, P. Vieu, Locally modelled regression and functional data, J. Nonparametr Stat., 22 (2010), 617–632. https://doi.org/10.1080/10485250903089930 doi: 10.1080/10485250903089930
[37]	L. Kara-Zaitri, A. Laksaci, M. Rachdi, P. Vieu, Data-driven kNN estimation in nonparametric functional data analysis, J. Multivariate Anal., 153 (2017), 176–188. https://doi.org/10.1016/j.jmva.2016.09.016 doi: 10.1016/j.jmva.2016.09.016
[38]	F. Ferraty, A. Laksaci, A. Tadj, P. Vieu, Rates of uniform consistency for nonparametric estimates with functional variables, J. Stat. Plan. Infer., 140 (2010), 335–352. https://doi.org/10.1016/j.jspi.2009.07.019 doi: 10.1016/j.jspi.2009.07.019
[39]	F. Burba, F. Ferraty, P. Vieu, k-Nearest Neighbour method in functional nonparametric regression, J. Nonparametr. Stat., 21 (2009), 453–469. https://doi.org/10.1080/10485250802668909 doi: 10.1080/10485250802668909
[40]	M. Rachdi, P. Vieu, Nonparametric regression for functional data: Automatic smoothing parameter selection, J. Statist. Plann. Inference, 137 (2007), 2784–2801. https://doi.org/10.1016/j.jspi.2006.10.001 doi: 10.1016/j.jspi.2006.10.001
[41]	G. Aneiros-Pérez, P. Vieu, Automatic estimation procedure in partial linear model with functional data, Stat. Papers, 52 (2011), 751–771. https://doi.org/10.1007/s00362-009-0280-2 doi: 10.1007/s00362-009-0280-2
[42]	H. L. Shang, Bayesian bandwidth estimation for a semi-functional partial linear regression model with unknown error density, Comput Stat., 29 (2014), 829–848. https://doi.org/10.1007/s00180-013-0463-0 doi: 10.1007/s00180-013-0463-0
[43]	G. Chagny, A. Roche, Adaptive estimation in the functional nonparametric regression model, J. Multivariate Anal., 146 (2016), 105–118. https://doi.org/10.1016/j.jmva.2015.07.001 doi: 10.1016/j.jmva.2015.07.001
[44]	L. Kara-Zaitri, A. Laksaci, M. Rachdi, P. Vieu, Uniform in bandwidth consistency for various kernel estimators involving functional data, J. Nonparametr. Stat., 29 (2017), 85–107. https://doi.org/10.1080/10485252.2016.1254780 doi: 10.1080/10485252.2016.1254780

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)