  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.
