Beyond black box densities: Parameter learning for the deviated components

As we collect additional samples from a data population for which a known density function estimate may have been previously obtained by a black box method, the increased complexity of the data set may result in the true density being deviated from the known estimate by a mixture distribution. Via a novel notion of distinguishability between the known density h_{0} and the deviated mixture distribution, we establish rates of convergence for the maximum likelihood estimates of \lambda {*} and G {*} under Wasserstein metric. Simulation studies are carried out to illustrate the theory.