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. To model this phenomenon, we consider the deviating mixture model (1-λ*)h0 + λ* (sumi = 1k pi* f(x|θi*)), where h0 is a known density function, while the deviated proportion λ* and latent mixing measure G* = sumi = 1k pi* δ_θi* associated with the mixture distribution are unknown. Via a novel notion of distinguishability between the known density h0 and the deviated mixture distribution, we establish rates of convergence for the maximum likelihood estimates of λ* and G* under Wasserstein metric. Simulation studies are carried out to illustrate the theory.