Estimation of a semiparametric contaminated regression model

We consider in this paper a contamined regression model where the distribution of the contaminating component is known when the Eu- clidean parameters of the regression model, the noise distribution, the contamination ratio and the distribution of the design data are un- known. Our model is said to be semiparametric in the sense that the probability density function (pdf) of the noise involved in the regression model is not supposed to belong to a parametric density family. When the pdf's of the noise and the contaminating phenomenon are supposed to be symmetric about zero, we propose an estimator of the various (Eu- clidean and functionnal) parameters of the model, and prove under mild conditions its convergence. We prove in particular that, under technical conditions all satisfied in the Gaussian case, the Euclidean part of the model is estimated at the rate $o_{a.s}(n-1/4+\gamma), $\gamma> 0$. We recall that, as it is pointed out in Bordes and Vandekerkhove (2010), this result cannot be ignored to go further in the asymptotic theory for this class of models. Finally the implementation and numerical performances of our method are discussed on several toy examples.