Convergence rate phase transitions of the moment estimator in finite Gaussian mixture models

This paper investigates the asymptotic behavior of the method of moments estimator in finite Gaussian mixture models, specifically in the over-specified setting where the true distribution is a single-component Gaussian nested within a two-component mixture model. The convergence rate of the estimator varies sharply with the local geometry of the parameter space. A spectrum of convergence rates is identified, ranging from the classical parametric rate n−1/2 to the minimax-optimal n−1/6 near singularities, depending on how the true parameters approach the boundary of identifiability. Although the n−1/6 rate has been previously established, prior work imposed restrictive boundedness assumptions and did not address the estimator’s behavior within small neighborhoods of degeneracy. A refined local analysis of the moment equations is developed to characterize the rate transition as parameters converge to the singular regime. The theoretical findings are supported by extensive simulations that provide the first empirical validation of these rate transitions and demonstrate their finite-sample impact. The results uncover surprisingly rich asymptotic behavior of moment matching near singularities. The analytical framework contributes to a deeper understanding of adaptive estimation, localized minimax theory, and inference under weak identifiability domains where substantial open questions remain.