Adaptive estimation for an inverse regression model with unknown operator

Abstract We are interested in the problem of estimating a regression function φ observed with a correlated noise Y  =  φ ( X )+ U . Contrary to the usual regression model, U is not centered conditionaly on X but rather on an observed variable W . Hence this model turns to be a difficult inverse problem where the corresponding operator is unknown since it is related to the joint distribution of ( X,W ). We focus on the case where the eigenvalues of the corresponding operator are observed with small perturbations and, using a well adapted spectral cut-off estimation procedure, we build a data driven estimates and derive an oracle inequality.