# New $M$-estimators in semi-parametric regression with errors in
variables

Research paper by **Cristina Butucea, Marie-Luce Taupin**

Indexed on: **19 Jun '08**Published on: **19 Jun '08**Published in: **Mathematics - Statistics**

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#### 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 $Z=X+\epsilon$
involving independent and unobserved random variables $X,\xi,\epsilon$ plus a
regression function $f_{\theta^0}$, known up to a finite dimensional
$\theta^0$. The common densities of the $X_i$'s and of the $\xi_i$'s are
unknown, whereas the distribution of $\epsilon$ is completely known. We aim at
estimating the parameter $\theta^0$ by using the observations
$(Y_1,Z_1),...,(Y_n,Z_n)$. We propose an estimation procedure based on the
least square criterion $\tilde{S}_{\theta^0,g}(\theta)=\m
athbb{E}_{\theta^0,g}[((Y-f_{\theta}(X))^2w(X)]$ where $w$ is a weight function
to be chosen. We propose an estimator and derive an upper bound for its risk
that depends on the smoothness of the errors density $p_{\epsilon}$ and on the
smoothness properties of $w(x)f_{\theta}(x)$. Furthermore, we give sufficient
conditions that ensure that the parametric rate of convergence is achieved. We
provide practical recipes for the choice of $w$ in the case of nonlinear
regression functions which are smooth on pieces allowing to gain in the order
of the rate of convergence, up to the parametric rate in some cases. We also
consider extensions of the estimation procedure, in particular, when a choice
of $w_{\theta}$ depending on $\theta$ would be more appropriate.