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

arXiv: https://arxiv.org/abs/2006.00704 · HTML: https://mlsanity.com/paper/2006.00704/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2020.

## Built on by (most cited first, 6 of 6)

- [Demystifying Softmax Gating Function in Gaussian Mixture of Experts](https://mlsanity.com/paper/2305.03288/) (2023)
- [Refined Convergence Rates for Maximum Likelihood Estimation under Finite Mixture Models](https://mlsanity.com/paper/2202.08786/) (2022)
- [Lower Bounds on the Total Variation Distance Between Mixtures of Two Gaussians](https://mlsanity.com/paper/2109.01064/) (2021)
- [Support Recovery in Mixture Models with Sparse Parameters](https://mlsanity.com/paper/2202.11940/) (2022)
- [Dendrograms of Mixing Measures for Softmax-Gated Gaussian Mixture of Experts: Consistency Without Model Sweeps](https://mlsanity.com/paper/2510.12744/) (2025)
- [Fidelity-Diversity Metrics for Text](https://mlsanity.com/paper/2607.04563/) (2026)

## Abstract (arXiv)

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.