We consider nonparametric estimation of probability measures for parameters in problems where only aggregate (population level) data are available. We summarize an existing computational method for the estimation problem which has been developed over the past several decades [24, 5, 12, 28, 16]. Theoretical results are presented which establish the existence and consistency of very general (ordinary, generalized and other) least squares estimates and estimators for the measure estimation problem with specific application to random PDEs.
Keywords: 34A55; 46S50; 62G07; 93E24; aggregate data; existence and approximation of estimators; individual data; inverse problems.