Nonparametric trigonometric orthogonal regression estimation

Nora Saadi; Smail Adjabi

doi:10.1016/j.crma.2016.02.013

Statistics

Nonparametric trigonometric orthogonal regression estimation

Presented by: the Editorial Board

Nora Saadi ¹; Smail Adjabi ¹

¹ LAMOS Laboratory, University of Bejaia, Algeria

Comptes Rendus. Mathématique, Volume 354 (2016) no. 8, pp. 851-858.

Abstract
Résumé

Eubank, Hart, and Speckman (1990) [2] have investigated the nonparametric trigonometric regression estimator. They assumed that the observation $x_{i}$ points satisfy $\int_{a}^{x_{i}} ψ (s) d s = \frac{(1 + i)}{n}$ , $i = 1, . . ., n$ , where $ψ \in L^{1} [a, b]$ is a density satisfying certain smoothness conditions, and in a work by E. Rafajłowicz (1987) [3], the observation points coincide with knots of numerical quadratures. The aim of the present work is to introduce a new estimator of the regression function based on trigonometric series, for fixed point designs different from the ones considered so far, under milder restrictions on the observation points. This seems to be important since it may be numerically difficult to determine exactly the points $x_{i}$ satisfying the recent condition or the knots of appropriate numerical quadratures, especially when their number is large.

Eubank, Hart et Speckman (1990) [2] ont étudié l'estimation non paramétrique de la fonction de régression par des séries trigonométriques. Ils ont supposé que les observations $x_{i}$ satisfont la condition $\int_{a}^{x_{i}} ψ (s) d s = \frac{(1 + i)}{n}$ , $i = 1, . . ., n$ , où $ψ \in L^{1} [a, b]$ est une densité vérifiant certaines conditions de régularité. Dans un travail de Rafajłowicz (1987) [3], les observations coincident avec les nœuds des fonctions numériques quadratiques. Ce travail a pour objectif d'introduire un nouvel estimateur de la fonction de régression basé sur un système trigonométrique. On supposera que les observations sont prises en des points équidistants, car il est difficile de déterminer numériquement avec précision les points $x_{i}$ satisfaisant aux précédentes conditions, spécialement quand le nombre d'observations est grand.

Received: 2015-07-01
Accepted: 2016-02-10
Published online: 2016-05-17

DOI: 10.1016/j.crma.2016.02.013

Author's affiliations:

Nora Saadi ¹; Smail Adjabi ¹

¹ LAMOS Laboratory, University of Bejaia, Algeria

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     title = {Nonparametric trigonometric orthogonal regression estimation},
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     pages = {851--858},
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     language = {en},
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Nora Saadi; Smail Adjabi. Nonparametric trigonometric orthogonal regression estimation. Comptes Rendus. Mathématique, Volume 354 (2016) no. 8, pp. 851-858. doi : 10.1016/j.crma.2016.02.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.02.013/

References
Cited by

[1] J. Engel A simple wavelet approach to nonparametric regression from recursive partitioning schemes, J. Multivar. Anal., Volume 49 (1994), pp. 242-254

[2] R.L. Eubank; J.D. Hart; P. Speckman Trigonometric series regression estimators with an application to partially linear models, J. Multivar. Anal., Volume 32 (1990), pp. 70-83

[3] E. Rafajłowicz Nonparametric orthogonal series estimators of regression: a class attaining the optimal convergence rate in L2, Stat. Probab. Lett., Volume 5 (1987), pp. 219-224

[4] E. Rafajłowicz Nonparametric least-squares estimation of a regression function, Statistics, Volume 19 (1988), pp. 349-358

[5] L. Rutkowski Orthogonal series estimates of a regression function with application in system identification (W. Grossmann; G.C. Pflug; W. Wertz, eds.), Probability and Statistical Inference, Reidel, Dordrecht, The Netherlands, 1982, pp. 343-347

[6] N. Saadi; S. Adjabi On the estimation of the probability density by trigonometric series, Commun. Stat., Theory Methods, Volume 38 (2009), pp. 3583-3595

[7] I.I. Sharapudinov On convergence of least-squares estimators, Mat. Zametki, Volume 53 (1993), pp. 131-143 (in Russian)

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