Beyond Black Box Densities: Parameter Learning for the Deviated Components
This work addresses incremental improvements in density estimation for statistical modeling, focusing on specific scenarios with known baseline densities.
The paper tackles the problem of modeling deviations from a known density estimate when new data introduces complexity, proposing a deviating mixture model and establishing convergence rates for parameter estimates under the Wasserstein metric.
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 \emph{deviating mixture model} $(1-λ^{*})h_0 + λ^{*} (\sum_{i = 1}^{k} p_{i}^{*} f(x|θ_{i}^{*}))$, where $h_0$ is a known density function, while the deviated proportion $λ^{*}$ and latent mixing measure $G_{*} = \sum_{i = 1}^{k} p_{i}^{*} δ_{θ_i^{*}}$ 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.