New M-estimators in semiparametric regression with errors in variables

Abstract : In the regression model with errors in variables, we observe $n$ i.i.d. copies of $(Y,Z)$ satisfying $Y=f_{\theta^0}(X)+\xi$ and $Z=X+\varepsilon$ involving independent and unobserved random variables $X,\xi,\varepsilon$ plus a regression function $f_{\theta^0}$, known up to some finite dimensional $\theta^0$. The common densities of the $X_i$'s and of the $\xi_i$'s are unknown whereas the distribution of $\varepsilon$ is completely known. We aim at estimating the parameter $\theta^0$ by using the observations $(Y_1,Z_1),\cdots, (Y_n,Z_n)$. We propose two estimation procedures based on the least square criterion $\tilde S_{\theta^0,g}(\theta)=\mathbb{E}_{\theta^0,g}[((Y-f_\theta(X))^2w(X)]$ where $w$ is some weight function, to be chosen. In the first estimation procedure, $w$ does not depend on $\theta$ and the distribution of the $\xi$'s is unknown. The second estimation procedure is based on $S_{\theta^0,g}(\theta)=\mathbb{E}_{\theta^0,g}[((Y-f_\theta(X))^2-\sigma_{\xi,2}^2)w_\theta(X)]$ where $w_\theta$ is positive weight function, to be chosen, and requires the knowledge of $\sigma_{\xi,2}^2=\mbox{Var}(\xi)$. In both cases, we propose two estimators and derive upper bounds for the risk of those estimators, depending on the smoothness of the errors density $p_\varepsilon$ and on the smoothness properties of $w(x)f_\theta(x)$ or $w_\theta(x)f_\theta(x)$ with respect to $x$. Furthermore we give sufficient conditions that ensure that the parametric rate of convergence is achieved. We provide practical recipes for the choice of $w$ or $ w_\theta$ in the case of nonlinear regressionfunctions which are smooth on pieces allowing to gain in the order of the rate of convergence, up to the parametric rate in some cases.
Type de document :
Article dans une revue
Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2008, 44 (3), pp.393-421
Liste complète des métadonnées


https://hal.archives-ouvertes.fr/hal-00013229
Contributeur : Cristina Butucea <>
Soumis le : vendredi 4 novembre 2005 - 13:31:31
Dernière modification le : jeudi 16 mars 2017 - 01:07:38
Document(s) archivé(s) le : lundi 20 septembre 2010 - 11:24:32

Identifiants

Collections

Citation

Cristina Butucea, Marie-Luce Taupin. New M-estimators in semiparametric regression with errors in variables. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institute Henri Poincaré, 2008, 44 (3), pp.393-421. <hal-00013229v2>

Partager

Métriques

Consultations de
la notice

283

Téléchargements du document

73