Beyond Black Box Densities: Parameter Learning for the Deviated Components
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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^*)h_0 + λ^* (∑_i = 1^k p_i^* f(x|θ_i^*)), where h_0 is a known density function, while the deviated proportion λ^* and latent mixing measure G_* = ∑_i = 1^k p_i^*δ_θ_i^* associated with the mixture distribution are unknown. 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 λ^* and G^* under Wasserstein metric. Simulation studies are carried out to illustrate the theory.
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