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) + ξ, Z = X + σε, involving independent and unobserved random variablesX, ξ, ε. The density g ofX is unknown, whereas the density of σε is completely known. Using the observations (Yi, Zi), i = 1, · · · , n, we propose an estimator of the regression function f , built as the ratio of two penalized minimum contrast estimators of l = fg and g, without any prior knowledge on their smoothness. We prove that its L2-risk on a compact set is bounded by the sum of the two L2(R)-risks of the estimators of l and g, and give the rate of convergence of such estimators for various smoothness classes for l and g, when the errors ε are either ordinary smooth or super smooth. The resulting rate is optimal in a minimax sense in all cases where lower bounds are available.