<i>k</i>NN local linear estimation of the conditional cumulative distribution function: Dependent functional data case

Ibrahim M. Almanjahie; Zouaoui Chikr Elmezouar; Ali Laksaci; Mustapha Rachdi

doi:10.1016/j.crma.2018.09.001

Statistics

kNN local linear estimation of the conditional cumulative distribution function: Dependent functional data case

Presented by: Paul Deheuvels

Ibrahim M. Almanjahie ¹; Zouaoui Chikr Elmezouar ¹; Ali Laksaci ¹; Mustapha Rachdi ²

¹ Department of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi Arabia
² Université Grenoble Alpes, Laboratoire AGEIS, EA 7407, AGIM–TIMB Team, UFR SHS, BP 47, 38040 Grenoble cedex 09, France

Comptes Rendus. Mathématique, Volume 356 (2018) no. 10, pp. 1036-1039.

Abstract
Résumé

Let ${(X_{i}, Y_{i})}_{i = 1, \dots, n}$ be a sequence of strongly mixing random vectors valued in $F \times R$ , where $F$ is a Hilbert space. This note deals with the problem of the local linear estimation of the conditional cumulative distribution function of $Y_{i}$ given $X_{i}$ . Then, the main goals of this note are (i) to construct a fast kNN local linear estimate of the conditional distribution function, and (ii) to prove its strong consistency in the functional time series analysis situation.

Soit ${(X_{i}, Y_{i})}_{i = 1, \dots, n}$ une suite de vecteurs aléatoires, fortement mélangeants, à valeurs dans $F \times R$ , où $F$ est un espace de Hilbert. Dans cette note, nous nous intéressons au problème d'estimation locale linéaire de la distribution conditionnelle de $Y_{i}$ sachant $X_{i}$ . Les objectifs principaux de cette note sont donc (i) de construire un estimateur local linéaire rapide, en introduisant la technique des k plus proches voisins, de la distribution conditionnelle et (ii) de prouver sa consistance forte dans le cadre des séries temporelles conditionnelles.

Received: 2018-06-11
Accepted: 2018-09-10
Published online: 2018-09-27

DOI: 10.1016/j.crma.2018.09.001

Author's affiliations:

Ibrahim M. Almanjahie ¹; Zouaoui Chikr Elmezouar ¹; Ali Laksaci ¹; Mustapha Rachdi ²

¹ Department of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi Arabia
² Université Grenoble Alpes, Laboratoire AGEIS, EA 7407, AGIM–TIMB Team, UFR SHS, BP 47, 38040 Grenoble cedex 09, France

@article{CRMATH_2018__356_10_1036_0,
     author = {Ibrahim M. Almanjahie and Zouaoui Chikr Elmezouar and Ali Laksaci and Mustapha Rachdi},
     title = {\protect\emph{k}NN local linear estimation of the conditional cumulative distribution function: {Dependent} functional data case},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1036--1039},
     publisher = {Elsevier},
     volume = {356},
     number = {10},
     year = {2018},
     doi = {10.1016/j.crma.2018.09.001},
     language = {en},
}

TY  - JOUR
AU  - Ibrahim M. Almanjahie
AU  - Zouaoui Chikr Elmezouar
AU  - Ali Laksaci
AU  - Mustapha Rachdi
TI  - kNN local linear estimation of the conditional cumulative distribution function: Dependent functional data case
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 1036
EP  - 1039
VL  - 356
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2018.09.001
LA  - en
ID  - CRMATH_2018__356_10_1036_0
ER  -

%0 Journal Article
%A Ibrahim M. Almanjahie
%A Zouaoui Chikr Elmezouar
%A Ali Laksaci
%A Mustapha Rachdi
%T kNN local linear estimation of the conditional cumulative distribution function: Dependent functional data case
%J Comptes Rendus. Mathématique
%D 2018
%P 1036-1039
%V 356
%N 10
%I Elsevier
%R 10.1016/j.crma.2018.09.001
%G en
%F CRMATH_2018__356_10_1036_0

Ibrahim M. Almanjahie; Zouaoui Chikr Elmezouar; Ali Laksaci; Mustapha Rachdi. kNN local linear estimation of the conditional cumulative distribution function: Dependent functional data case. Comptes Rendus. Mathématique, Volume 356 (2018) no. 10, pp. 1036-1039. doi : 10.1016/j.crma.2018.09.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.09.001/

References
Cited by

[1] G. Aneiros; E.G. Bongiorno; R. Cao; P. Vieu Functional Statistics and Related Fields, Contributions to Statistics, Springer International Publishing, 2017

[2] A. Baìllo; A. Grané Local linear regression for functional predictor and scalar response, J. Multivar. Anal., Volume 100 (2009), pp. 102-111

[3] J. Barrientos-Marin; F. Ferraty; P. Vieu Locally modelled regression and functional data, J. Nonparametr. Stat., Volume 22 (2010) no. 5, pp. 617-632

[4] A. Berlinet; A. Elamine; A. Mas Local linear regression for functional data, Ann. Inst. Stat. Math., Volume 63 (2011), pp. 1047-1075

[5] F. Burba; F. Ferraty; P. Vieu k-nearest neighbor method in functional non-parametric regression, J. Nonparametr. Stat., Volume 21 (2009), pp. 453-469

[6] G. Collomb Estimation non paramétrique de la régression : Revue bibliographique, Int. Stat. Rev., Volume 49 (1981), pp. 75-93

[7] A. Cuevas A partial overview of the theory of statistics with functional data, J. Stat. Plan. Inference, Volume 147 (2014), pp. 1-23

[8] J. Demongeot; A. Laksaci; M. Rachdi; S. Rahmani On the local linear modelization of the conditional distribution for functional data, Sankhya, Ser. A, Volume 76 (2014), pp. 328-355

[9] J. Fan; I. Gijbels Local Polynomial Modelling and Its Applications, Chapman & Hall, London, 1996

[10] F. Ferraty; P. Vieu Nonparametric Functional Data Analysis. Theory and Practice, Springer Series in Statistics, Springer, New York, 2006

[11] G. Geenens Curse of dimensionality and related issues in nonparametric functional regression, Stat. Surv., Volume 5 (2011), pp. 30-43

[12] A. Goia; P. Vieu An introduction to recent advances in high/infinite dimensional statistics, J. Multivar. Anal., Volume 146 (2016), pp. 1-6

[13] L. Kara-Zaitri; A. Laksaci; M. Rachdi; P. Vieu Data-driven kNN estimation for various problems involving functional data, J. Multivar. Anal., Volume 153 (2017), pp. 176-188

[14] N. Kudraszow; P. Vieu Uniform consistency of kNN regressors for functional variables, Stat. Probab. Lett., Volume 83 (2013), pp. 1863-1870

[15] A. Laksaci; M. Rachdi; S. Rahmani Spatial modelization: local linear estimation of the conditional distribution for functional data, Spatial Statist., Volume 6 (2013), pp. 1-23

[16] T. Laloë A k-nearest neighbor approach for functional regression, Stat. Probab. Lett., Volume 78 (2008), pp. 1189-1193

[17] H. Lian Convergence of functional k-nearest neighbor regression estimate with functional responses, Electron. J. Stat., Volume 5 (2011), pp. 31-40

[18] N. Ling; P. Vieu Nonparametric modelling for functional data: selected survey and tracks for future, Statistics, Volume 52 (2018), pp. 934-949

[19] E. Masry Recursive probability density estimation for weakly dependent stationary processes, IEEE Trans. Inf. Theory, Volume 32 (1986), pp. 254-267

[20] Z. Zhou; Z. Lin Asymptotic normality of locally modelled regression estimator for functional data, J. Nonparametr. Stat., Volume 28 (2016), pp. 116-131

Cited by Sources:

Comments - Policy