The nonparametric estimation of a regression function from conditional moment restrictions involving instrumental variables is considered. The rate of convergence of penalized estimators is studied in the case where the regression function is not identified from the conditional moment...

This paper studies the estimation of a nonparametric function <italic>ϕ</italic> from the inverse problem <italic>r</italic> = <italic>Tϕ</italic> given estimates of the function <italic>r</italic> and of the linear transform <italic>T</italic>. We show that rates of convergence of the estimator are driven by two types of assumptions expressed in a single Hilbert scale. The...