may be required for well-posedness. 

Well-posedness is important for plug-in estimation since if the estimators are in a class where the problem is well-posed they are consistent, and con- versely, if well-posedness does not hold consistency will fail for some cases. Lack of well-posedness can be remedied by regularization, but the price is often more extensive requirements on the model and slower convergence. For example, in deconvolution (see e.g. Fan, 1991, and most other papers cited here) spectral cut-off regularization is utilized; it crucially depends on know- ing the rate of the decay at infinity of the density. 

Often non-parametric identification is used to justify parametric or semi- parametric estimation; the claim here is that well-posedness should be an important part of this justification. The reason for that is that in estimat- ing a possibly misspecified parametric model, the misspecified functions of the observables belong in a nonparametric neighborhood of the true func- tions; if the model is non-parametrically identified, the unique solution to the true model exists, but without well-posedness the solution to the para- metric model and to the true one may be far apart. 

For deconvolution An and Hu (2012) demonstrate well-posedness in spaces of integrable density functions when the measurement error has a mass point; this may happen in surveys when probability of truthful reporting is non- zero. The conditions for well-posedness here are provided in S $^{∗}$; this then additionally does not exclude mass points in the distribution of the mis- measured variable itself; there is some empirical evidence of mass points in 

