Adaptive estimation for an inverse regression model with unknown operator

Abstract

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.