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On the nonparametric estimation of the functional expectile regression
[Sur l’estimation non-paramétrique dans un modèle de régression expectile fonctionnelle]
Comptes Rendus. Mathématique, Tome 358 (2020) no. 3, pp. 267-272.

Dans cette note, nous nous intéressons au problème d’estimation non-paramétrique de la fonction de régression expectile lorsqu’on régresse une variable réelle sur une variable fonctionnelle. Plus précisément, nous obtenons la convergence presque complète de l’estimateur à noyau de la fonction de régression expectile sous des conditions générales. Nous discutons brièvement notre résultat et mettons en évidence le lien avec la fonction de régression.

In this note, we investigate the kernel-type estimator of the nonparametric expectile regression model for functional data. More precisely, we establish the almost complete convergence rate of this estimator under some mild conditions. Finally, the usefulness of the expectile regression is discussed, in particular, the connection with the regression function.

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DOI : https://doi.org/10.5802/crmath.27
@article{CRMATH_2020__358_3_267_0,
     author = {Mustapha Mohammedi and Salim Bouzebda and Ali Laksaci},
     title = {On the nonparametric estimation of the functional expectile regression},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {267--272},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {3},
     year = {2020},
     doi = {10.5802/crmath.27},
     language = {en},
}
Mustapha Mohammedi; Salim Bouzebda; Ali Laksaci. On the nonparametric estimation of the functional expectile regression. Comptes Rendus. Mathématique, Tome 358 (2020) no. 3, pp. 267-272. doi : 10.5802/crmath.27. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.27/

[1] Belkacem Abdous; Bruno Remillard Relating quantiles and expectiles under weighted-symmetry, Ann. Inst. Stat. Math., Volume 47 (1995) no. 2, pp. 371-384 | Article | MR 1345429 | Zbl 0833.62013

[2] Dennis J. Aigner; Takeshi Amemiya; Dale J. Poirier On the estimation of production frontiers: maximum likelihood estimation of the parameters of a discontinuous density function, Int. Econ. Rev., Volume 17 (1976), pp. 377-396 | Article | MR 428614 | Zbl 0339.62083

[3] Fahimah A. Al-Awadhi; Zoulikha Kaid; Ali Laksaci; Idir Ouassou; Mustapha Rachdi Functional data analysis: local linear estimation of the L 1 -conditional quantiles, Stat. Methods Appl., Volume 28 (2019) no. 2, pp. 217-240 | Article | MR 3954406 | Zbl 1427.62146

[4] Functional Statistics and Related Fields (Corunna, Spain, June 15–17, 2017) (Germán Aneiros; Enea G. Bongiorno; Ricardo Cao; Philippe Vieu, eds.), Contributions to Statistics, Springer, 2017 | Zbl 1373.62016

[5] Fabio Bellini; E. D. Bernardino Risk management with expectiles, Eur. J. Finance, Volume 23 (2017) no. 6, pp. 487-506 | Article

[6] Fabio Bellini; Valeria Bignozzi; Giovanni Puccetti Conditional expectiles, time consistency and mixture convexity properties, Insur. Math. Econ., Volume 82 (2018), pp. 117-123 | Article | MR 3850612 | Zbl 1416.91156

[7] Abdelaati Daouia; Irène Gijbels; Gilles Stupfler Extremiles: A New Perspective on Asymmetric Least Squares, J. Am. Stat. Assoc., Volume 114 (2019) no. 527, pp. 1366-1381 | Article | MR 4011785 | Zbl 1428.62198

[8] Abdelaati Daouia; Davy Paindaveine From halfspace m-depth to multiple-output expectile regression (2019) (https://arxiv.org/abs/1905.12718)

[9] Bradley Efron Regression percentiles using asymmetric squared error loss, Stat. Sin., Volume 1 (1991) no. 1, pp. 93-125 | MR 1101317 | Zbl 0822.62054

[10] Werner Ehm; Tilmann Gneiting; Alexander Jordan; Fabian Krüger Of quantiles and expectiles: consistent scoring functions, Choquet representations, and forecast rankings, J. R. Stat. Soc., Ser. B, Stat. Methodol., Volume 78 (2016) no. 3, pp. 505-562 | MR 3506792 | Zbl 1414.62038

[11] Frédéric Ferraty; Philippe Vieu Nonparametric functional data analysis. Theory and Practice, Springer Series in Statistics, Springer, 2006 | Zbl 1119.62046

[12] Michael C. Jones Expectiles and M-quantiles are quantiles, Stat. Probab. Lett., Volume 20 (1994) no. 2, pp. 149-153 | Article | MR 1293293 | Zbl 0801.62012

[13] Roger Koenker; Gilbert jun. Bassett Regression quantiles, Econometrica, Volume 46 (1978), pp. 33-50 | Article | MR 474644 | Zbl 0373.62038

[14] Chung-Ming Kuan; Jin-Huei Yeh; Yu-Chin Hsu Assessing value at risk with CARE, the Conditional Autoregressive Expectile models, J. Econom., Volume 150 (2009) no. 2, pp. 261-270 | Article | MR 2535521 | Zbl 1429.62474

[15] Ali Laksaci; Mohamed Lemdani; Elias Ould-Saïd A generalized L 1 -approach for a kernel estimator of conditional quantile with functional regressors: consistency and asymptotic normality, Stat. Probab. Lett., Volume 79 (2009) no. 8, pp. 1065-1073 | Article | MR 2510768 | Zbl 1158.62318

[16] Ali Laksaci; Mohamed Lemdani; Elias Ould-Saïd Asymptotic results for an L 1 -norm kernel estimator of the conditional quantile for functional dependent data with application to climatology, Sankhyā, Ser. A, Volume 73 (2011) no. 1, pp. 125-141 | Article | MR 2887090 | Zbl 1267.62050

[17] Véronique Maume-Deschamps; Didier Rullière; Khalil Said Multivariate extensions of expectiles risk measures, Depend. Model., Volume 5 (2017), pp. 20-44 | Article | MR 3619110 | Zbl 1358.91113

[18] Whitney K. Newey; James L. Powell Asymmetric least squares estimation and testing, Econometrica, Volume 55 (1987), pp. 819-847 | Article | MR 906565 | Zbl 0625.62047

[19] James O. Ramsay; Bernard W. Silverman Functional Data Analysis, Springer Series in Statistics, Springer, 2005 | Zbl 1079.62006

[20] J. W. Taylor Estimating Value at Risk and Expected Shortfall Using Expectiles, J. Financial Econom., Volume 6 (2008), pp. 231-252 | Article

[21] Linda Schulze Waltrup; Fabian Sobotka; Thomas Kneib; Goran Kauermann Expectile and quantile regression—David and Goliath?, Stat. Model., Volume 15 (2015) no. 5, pp. 433-456 | Article | MR 3403125