Functional single-index $ k $-nearest neighbor relative error regression under weak dependence

Fatimah Alshahrani; Wahiba Bouabsa; Fatimah Alshahrani; Wahiba Bouabsa

doi:10.3934/math.2026779

AIMS Mathematics

2026, Volume 11, Issue 6: 19127-19161. doi: 10.3934/math.2026779

Previous Article Next Article

Research article

Functional single-index $ k $-nearest neighbor relative error regression under weak dependence

Fatimah Alshahrani ^{1
,},
Wahiba Bouabsa ^{2
,
,}

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia; fmalshahrani@pnu.edu.sa
2.
Laboratory of Statistics and Stochastic Processes, University of Djillali Liabes BP 89, Sidi Bel Abbes 22000, Algeria

Received: 24 April 2026 Revised: 09 June 2026 Accepted: 12 June 2026 Published: 29 June 2026
MSC : 62G07, 62G20, 62G35, 62H12

We study estimation of the relative error regression function in a functional single-index regression model under quasi-associated dependence. We introduce a $ k $-nearest neighbors ($ k $-NN) estimator whose smoothing adapts to the local concentration of the covariate trajectories. Almost complete convergence rates are established under mild small-ball and dependence conditions. A simulation study compares the proposed $ k $-NN estimator with its kernel counterpart, and a real data application to air quality forecasting (NOx $ \rightarrow $ ozone) illustrates it's practical performance.
- functional single-index model,
- relative error regression,
- $ k $-nearest neighbors,
- kernel estimator,
- almost complete convergence,
- quasi-associated dependence
Citation: Fatimah Alshahrani, Wahiba Bouabsa. Functional single-index $ k $-nearest neighbor relative error regression under weak dependence[J]. AIMS Mathematics, 2026, 11(6): 19127-19161. doi: 10.3934/math.2026779

Related Papers:

Abstract

We study estimation of the relative error regression function in a functional single-index regression model under quasi-associated dependence. We introduce a $ k $-nearest neighbors ($ k $-NN) estimator whose smoothing adapts to the local concentration of the covariate trajectories. Almost complete convergence rates are established under mild small-ball and dependence conditions. A simulation study compares the proposed $ k $-NN estimator with its kernel counterpart, and a real data application to air quality forecasting (NOx $ \rightarrow $ ozone) illustrates it's practical performance.

References

[1]	D. Bosq, Linear processes in function spaces: theory and applications, Springer, 2000. https://doi.org/10.1007/978-1-4612-1154-9
[2]	J. Ramsay, B. Silverman, Functional data analysis, 2 Eds., Springer, 2005. https://doi.org/10.1007/b98888
[3]	F. Ferraty, P. Vieu, Nonparametric functional data analysis: theory and practice, Springer, 2006. https://doi.org/10.1007/0-387-36620-2
[4]	L. Horvát, P. Kokoszka, Inference for functional data with applications, Springer, 2012. https://doi.org/10.1007/978-1-4614-3655-3
[5]	T. Hsing, R. Eubank, Theoretical foundations of functional data analysis, John Wiley & Sons, Ltd, 2015. https://doi.org/10.1002/9781118762547
[6]	G. Aneiros, E. G. Bongiorno, R. Cao, P. Vieu, Functional statistics and related fields, Springer, 2017. https://doi.org/10.1007/978-3-319-55846-2
[7]	J. D. Esary, F. Proschan, D. W. Walkup, Association of random variables, with applications, Ann. Math. Stat., 38 (1967), 1466–1474. https://doi.org/10.1214/aoms/1177698701 doi: 10.1214/aoms/1177698701
[8]	F. Alshahrani, W. Bouabsa, I. M. Almanjahie, M. K. Attouch, $k$NN 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
[9]	F. Alshahrani, W. Bouabsa, I. M. Almanjahie, M. K. Attouch, Robust kernel regression function with uncertain scale parameter for high dimensional ergodic data using $k$-nearest neighbor estimation, AIMS Math., 8 (2023), 13000–13023. https://doi.org/10.3934/math.2023655 doi: 10.3934/math.2023655
[10]	W. Härdle, P. Hall, H. Ichimura, Optimal smoothing in single-index models, Ann. Stat., 21 (1993), 157–178. https://doi.org/10.1214/aos/1176349020 doi: 10.1214/aos/1176349020
[11]	M. Hristache, A. Juditsky, V. Spokoiny, Direct estimation of the index coefficient in a single-index model, Ann. Stat., 29 (2001), 595–623. https://doi.org/10.1214/aos/1009210682 doi: 10.1214/aos/1009210682
[12]	F. Ferraty, A. Peuch, P. Vieu, Modèle à indice fonctionnel simple single functional index model, C. R. Math., 336 (2003), 1025–1028. https://doi.org/10.1016/S1631-073X(03)00239-5 doi: 10.1016/S1631-073X(03)00239-5
[13]	F. Ferraty, J. Park, P. Vieu, Estimation of a functional single index model, In: F. Ferraty, Recent advances in functional data analysis and related topics, 2011,111–116. https://doi.org/10.1007/978-3-7908-2736-1_17
[14]	M. C. Jones, H. Park, K. I. Shin, S. K. Vines, S. O. Jeong, Relative error prediction via kernel regression smoothers, J. Stat. Plann. Inference, 138 (2008), 2887–2898. https://doi.org/10.1016/j.jspi.2007.11.001 doi: 10.1016/j.jspi.2007.11.001
[15]	I. Almanjahie, K. Aissiri, A. Laksaci, Z. Elmezouar, The $k$ nearest neighbors smoothing of the relative error regression with functional regressor, Commun. Stat., 51 (2022), 4196–4209. https://doi.org/10.1080/03610926.2020.1811870 doi: 10.1080/03610926.2020.1811870
[16]	T. Laloë, A $k$-nearest neighbor approach for functional regression, Stat. Probab. Lett., 78 (2008), 1189–1193. https://doi.org/10.1016/j.spl.2007.11.014 doi: 10.1016/j.spl.2007.11.014
[17]	F. Burba, F. Ferraty, P. Vieu, $k$-nearest neighbour method in functional regression, J. Nonparametric Stat., 21 (2009), 453–469. https://doi.org/10.1080/10485250802668909 doi: 10.1080/10485250802668909
[18]	H. Lian, Convergence of functional $k$-nearest neighbor regression estimate with functional responses, Electron. J. Stat., 5 (2011), 31–40. https://doi.org/10.1214/11-EJS595 doi: 10.1214/11-EJS595
[19]	N. L. Kudraszow, P. Vieu, Uniform consistency of $k$NN regressors for functional variables, Stat. Probab. Lett., 83 (2013), 1863–1870. https://doi.org/10.1016/j.spl.2013.04.017 doi: 10.1016/j.spl.2013.04.017
[20]	A. Bulinski, C. Suquet, Normal approximation for quasi-associated random fields, Stat. Probab. Lett., 54 (2001), 215–226. https://doi.org/10.1016/S0167-7152(01)00108-0 doi: 10.1016/S0167-7152(01)00108-0
[21]	L. Douge, Théorèmes limites pour des variables quasi-associées hilbertiennes, Ann. I'ISUP, 54 (2010), 51–60.
[22]	M. Ezzahrioui, E. Ould-Saïd, Asymptotic normality of a nonparametric estimator of the conditional mode function for functional data, J. Nonparametric Stat., 20 (2008), 3–18. https://doi.org/10.1080/10485250701541454 doi: 10.1080/10485250701541454
[23]	P. Doukhan, S. Louhichi, A new weak dependence condition and applications to moment inequalities, Stochastic Processes Appl., 84 (1999), 313–342. https://doi.org/10.1016/S0304-4149(99)00055-1 doi: 10.1016/S0304-4149(99)00055-1
[24]	M. K. Attouch, W. Bouabsa, The $k$-nearest neighbors estimation of the conditional mode for functional data, Rev. Roumaine Math. Pures Appl., 58 (2013), 393–415.
[25]	A. Ait-Saïdi, F. Ferraty, R. Kassa, P. Vieu, Cross-validation in functional single-index regression, Statistics, 42 (2008), 475–494. https://doi.org/10.1080/02331880801980377 doi: 10.1080/02331880801980377
[26]	R. S. Kallabis, M. H. Neumann, An exponential inequality under weak dependence, Bernoulli, 12 (2006), 333–350. https://doi.org/10.3150/bj/1145993977 doi: 10.3150/bj/1145993977
[27]	L. Douge, Vitesses de convergence dans la loi forte des grands nombres et dans l'estimation de la densité pour des variables aléatoires associées, C. R. Math., 344 (2007), 515–518. https://doi.org/10.1016/j.crma.2007.02.017 doi: 10.1016/j.crma.2007.02.017

Reader Comments

Your name:*

Email:*
© 2026 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)