Before revision: This paper studies non-identified ill-posed inverse models with estimated operator. Leading examples are the nonparametric IV regression and the functional linear IV regression. We argue that identification of infinite-dimensional parameters is less crucial than identification of finite-dimensional parameters. We show that in the case of identification failures , a very general family of continuously-regularized estimators is consistent for the best approximation of the parameter of interest and obtain L_2 and L_\infty finite-sample risk bounds.
After revision: This paper studies non-identified ill-posed inverse models. The Tikhonov-regularized estimator converges to a well-defined limit equal to the best approximation of the parameter of interest and obtain L_2 and L_\infty finite-sample risk bounds.