A new spatial regression estimator in the multivariate context

Dabo-Niang, Sophie; Ternynck, Camille; Yao, Anne-Francoise

Comptes Rendus. Mathématique, Tome 353 (2015) no. 7, pp. 635-639.

Dans cette note, nous proposons un estimateur non paramétrique spatial de la fonction de régression $x\to r(x):=\mathbb{E}[Y_{\mathbf{i}}|X_{\mathbf{i}}=x],x\in {\mathbb{R}}^d$, d'un champ stationnaire $\{(Y_{\mathbf{i}},X_{\mathbf{i}}),\mathbf{i}\in {\mathbb{Z}}^N\}$ de dimension $(d+1)$, à un point localisé à un site donné j. L'estimateur proposé est composé de deux noyaux permettant de contrôler à la fois la distance entre les observations et entre les sites. La convergence presque complète ainsi que la convergence en moyenne d'ordre q (norme $L^q$) $(q\in {\mathbb{N}}^{⁎})$ de l'estimateur à noyaux sont obtenus en considérant des processus α-mélangeants.

In this note, we propose a nonparametric spatial estimator of the regression function $x\to r(x):=\mathbb{E}[Y_{\mathbf{i}}|X_{\mathbf{i}}=x],x\in {\mathbb{R}}^d$, of a stationary $(d+1)$-dimensional spatial process $\{(Y_{\mathbf{i}},X_{\mathbf{i}}),\mathbf{i}\in {\mathbb{Z}}^N\}$, at a point located at some station j. The proposed estimator depends on two kernels in order to control both the distance between observations and the spatial locations. Almost complete convergence and consistency in $L^q$ norm $(q\in {\mathbb{N}}^{⁎})$ of the kernel estimate are obtained when the sample considered is an α-mixing sequence.

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