On nonparametric maximum likelihood for a class of stochastic inverse problems

We establish the consistency of a nonparametric maximum likelihood estimator for a class of stochastic inverse problems. We proceed by embedding the framework into the general settings of early results of Pfanzagl related to mixtures [Pfanzagl, J., 1998a. Consistency of maximum likelihood estimators for certain nonparametric families, in particular: mixtures. J. Statist. Plann. Inference 19(2), 137-158, MR 89g:62063; Pfanzagl, J., 1998b. Large deviation inequality for maximum likelihood estimators for certain nonparametric families, in particular: mixtures. Ann. Stats. 19(2), 137-158, MR 89g:62063].