Uniform Convergence Rates for Maximum Likelihood Estimation under Two-Component Gaussian Mixture Models
This work addresses parameter estimation challenges in statistical modeling for researchers, but it is incremental as it builds on existing mixture model theory without introducing new methods.
The paper tackles the problem of deriving uniform convergence rates for maximum likelihood estimation in two-component Gaussian mixture models with unequal variances, showing a phase transition in optimal rates based on mixture balance, with theoretical rates supported by simulation studies.
We derive uniform convergence rates for the maximum likelihood estimator and minimax lower bounds for parameter estimation in two-component location-scale Gaussian mixture models with unequal variances. We assume the mixing proportions of the mixture are known and fixed, but make no separation assumption on the underlying mixture components. A phase transition is shown to exist in the optimal parameter estimation rate, depending on whether or not the mixture is balanced. Key to our analysis is a careful study of the dependence between the parameters of location-scale Gaussian mixture models, as captured through systems of polynomial equalities and inequalities whose solution set drives the rates we obtain. A simulation study illustrates the theoretical findings of this work.