Estimation non paramétrique de quantiles conditionnels pour des variables fonctionnelles spatialement dépendantes

Laksaci, Ali; Maref, Fouzia

Comptes Rendus. Mathématique, Tome 347 (2009) no. 17-18, pp. 1075-1080.

Étant donné un champ aléatoire fonctionnel, stationnaire ($Z_{\mathbf{i}}={(X_{\mathbf{i}},Y_{\mathbf{i}})}_{,\mathbf{i}\in {\mathbb{N}}^N}$, $N>0$) à valeurs dans $\mathcal{F}\times \mathbb{R}$, où $\mathcal{F}$ est un espace semi-métrique, de dimension éventuellement infinie. Dans cette Note, on se propose d'étudier la covariation spatiale des deux variables $X_{\mathbf{i}}$ et $Y_{\mathbf{i}}$ via l'estimation non paramétrique des quantiles conditionnels de $Y_{\mathbf{i}}$ sachant $X_{\mathbf{i}}$. Nous construisons un estimateur à noyau pour ce modèle non paramétrique spatial et nous établissons sa vitesse de convergence presque complètement.

Consider $Z_{\mathbf{i}}=(X_{\mathbf{i}},Y_{\mathbf{i}})$, $\mathbf{i}\in {\mathbb{N}}^N$ be a $\mathcal{F}\times \mathbb{R}$-valued measurable strictly stationary spatial process, where $\mathcal{F}$ is a semi-metric space. We study a kernel estimator of conditional quantiles of the univariate response variable $Y_{\mathbf{i}}$ given the functional variable $X_{\mathbf{i}}$. The main aim of this Note is to prove the almost complete convergence (with rate) of this estimate.

[1] Anselin, L.; Florax, R.J.G.M. New Directions in Spatial Econometrics, Springer, Berlin, 1995

[2] Carbon, M.; Francq, C.; Tran, L.T. Kernel regression estimation for random fields, J. Statist. Plann. Inference, Volume 137 (2007), pp. 778-798

[3] Cressie, N.A. Statistics for Spatial Data, Wiley, New York, 1991

[4] R.M. Crujeiras, W.G. Manteiga, A. Laksaci, E. Ould Saïd, Asymptotic properties for an $L^1$-norm kernel estimator of the spatial quantile regression for functional data. Technical report, no. 393, Mars 2009, LMPA, Université du Littoral Cote d'Opale, submitted for publication

[5] Dabo-Niang, S.; Yao, A.F. Kernel regression estimation for continuous spatial processes, Math. Methods Statist., Volume 16 (2007), pp. 298-317

[6] S. Dabo-Niang, A. Laksaci, Spatial conditional quantile regression: Weak consistency of a kernel estimate, 2009, submitted for publication

[7] Ezzahrioui, M.; Ould-Said, E. Asymptotic results of the kernel estimator of the conditional quantile in the normed space under α-mixing hypothesis, Comm. Statist. Theory Methods, Volume 37 (2008), pp. 2735-2759

[8] Ferraty, F.; Vieu, Ph. Nonparametric Functional Data Analysis, Springer Series in Statistics, Springer, New York, 2006

[9] Ferraty, F.; Rabhi, A.; Vieu, Ph. Conditional quantiles for functionally dependent data with application to the climatic El Nino Phenomenon, Sankhyia, Volume 67 (2005), pp. 378-399

[10] Ferraty, F.; Laksaci, A.; Vieu, Ph. Estimating some characteristics of the conditional distribution in nonparametric functional models, Stat. Inference Stoch. Process., Volume 9 (2006), pp. 47-76

[11] Guyon, X. Random Fields on a Network – Modeling, Statistics, and Applications, Springer, New York, 1995

[12] M. Hallin, Z. Lu, K. Yu, Local linear spatial quantile regression, Bernoulli (2009), in press

[13] A. Laksaci, E. Ould Saïd, Kernel estimator for the spatial regression quantile : $L_1$-approach. Technical report, no. 392, Mars 2009, LMPA, Université du Littoral Côte d'Opale, submitted for publication

[14] Laksaci, A.; Lemdani, M.; Ould-Saïd, E. A generalized $L^1$-approach for a kernel estimator of conditional quantile with functional regressors: Consistency and asymptotic normality, Statist. Probab. Lett., Volume 79 (2009), pp. 1065-1073

[15] Lemdani, M.; Ould-Saïd, E.; Poulin, N. Asymptotic properties of a conditional quantile estimator with randomly truncated data, J. Multivariate Anal., Volume 100 (2009), pp. 546-559

[16] Li, J.; Tran, L.T. Nonparametric estimation of conditional expectation, J. Statist. Plann. Inference, Volume 139 (2009), pp. 164-175

[17] Ripley, B. Spatial Statistics, Wiley, New York, 1981

[18] Samanta, M. Non-parametric estimation of conditional quantiles, Statist. Probab. Lett., Volume 7 (1989), pp. 407-412

[19] J Stone, C. Consistent nonparametric regression, Discuss. Ann. Statist., Volume 5 (1977), pp. 595-645

[20] Stute, W. Conditional empirical processes, Ann. Statist., Volume 14 (1986), pp. 638-647

[21] Tran, L.T. Kernel density estimation on random fields, J. Multivariate Anal., Volume 34 (1990), pp. 37-53

[22] E. Youndjé, Estimation non paramétrique de la densité conditionnelle par la méthode du noyau, Thèse de Doctarat, Université de Rouen, 1993