This paper studies nonparametric estimation of conditional moment models in which the residual functions could be nonsmooth with respect to the unknown functions of endogenous variables. It is a problem of nonparametric nonlinear instrumental variables (IV) estimation, and a difficult nonlinear ill-posed inverse problem with an unknown operator. We first propose a penalized sieve minimum distan...

In many applications of solving ill-posed problems there are a number of regularization methods to choose from as well as a free regularization parameter. The choice of the method and regularization can add bias if deblurred images are based on what the researcher expects. The presented project develops a tool for method choice and parameter selection by using a set of three statistical diagnos...

We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is ill-posed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions. This feature is modeled by a piecewi...

The characterization of the local ill–posedness and the local degree of nonlinearity are of particular importance for the stable solution of nonlinear ill–posed problems. We present assertions concerning the interdependence between the illposedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity combined with source conditions can...

We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is ill-posed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions. This feature is modeled by a piecewi...