# Nonparametric Estimation of the Regression Function in an Errors-in-Variables Model

Abstract : We consider the regression model with errors-in-variables where we observe $n$ i.i.d. copies of $(Y,Z)$ satisfying $Y=f(X)+\xi, \; Z=X+\sigma\varepsilon$, involving independent and unobserved random variables $X,\xi,\varepsilon$. The density $g$ of $X$ is unknown, whereas the density of $\sigma\varepsilon$ is completely known. Using the observations $(Y_i, Z_i)$, $i=1,\cdots,n$, we propose an estimator of the regression function $f$, built as the ratio of two penalized minimum contrast estimators of $\ell=fg$ and $g$, without any prior knowledge on their smoothness. We prove that its $\mathbb{L}_2$-risk on a compact set is bounded by the sum of the two $\mathbb{L}_2(\mathbb{R})$-risks of the estimators of $\ell$ and $g$, and give the rate of convergence of such estimators for various smoothness classes for $\ell$ and $g$, when the errors $\varepsilon$ are either ordinary smooth or super smooth. The resulting rate is optimal in a minimax sense in all cases where lower bounds are available.
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Article dans une revue
Statistica Sinica, Taipei : Institute of Statistical Science, Academia Sinica, 2007, 17, pp.1065-1090
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https://hal.archives-ouvertes.fr/hal-00013248
Contributeur : Marie-Luce Taupin <>
Soumis le : vendredi 4 novembre 2005 - 14:24:06
Dernière modification le : mardi 10 octobre 2017 - 13:34:41
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### Citation

Fabienne Comte, Marie-Luce Taupin. Nonparametric Estimation of the Regression Function in an Errors-in-Variables Model. Statistica Sinica, Taipei : Institute of Statistical Science, Academia Sinica, 2007, 17, pp.1065-1090. 〈hal-00013248〉

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