Uniform Convergence Rates for Maximum Likelihood Estimation under Two-Component Gaussian Mixture Models

Finite mixture models are a widely-used tool for modeling heterogeneous data, consisting of hidden subpopulations with distinct distributions. For applications exhibiting continuous data, location-scale Gaussian mixtures are arguably the most popular family of parametric mixture models. Beyond their broad applications as a modeling and clustering tool in the social, physical and life sciences (McLachlan & Peel 2004), Gaussian mixtures provide a flexible approach to density estimation (Genovese & Wasserman 2000, Ghosal & van der Vaart 2001). Estimating the parameters of a mixture model is crucial for quantifying the underlying heterogeneity of the data. One of the most widely-used approaches is the maximum likelihood estimator (MLE). A Gaussian mixture model with a known number of components K, all of which are well-separated, forms a regular parametric model for which the MLE achieves the standard parametric estimation rate (Ho & Nguyen 2016b, Chen 2017). Such rates are typically understood in terms of convergence of mixing measures, quantified using the Wasserstein distance as a means of avoiding label switching issues inherent in mixture modeling (Nguyen 2013).