# Singularity structures and impacts on parameter estimation in finite mixtures of distributions

arXiv: https://arxiv.org/abs/1609.02655 · HTML: https://mlsanity.com/paper/1609.02655/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2016.

## Built on by (most cited first, 9 of 9)

- [Convergence Rates for Gaussian Mixtures of Experts](https://mlsanity.com/paper/1907.04377/) (2019)
- [Towards Convergence Rates for Parameter Estimation in Gaussian-gated Mixture of Experts](https://mlsanity.com/paper/2305.07572/) (2023)
- [Strong identifiability and parameter learning in regression with heterogeneous response](https://mlsanity.com/paper/2212.04091/) (2022)
- [Uniform Consistency in Nonparametric Mixture Models](https://mlsanity.com/paper/2108.14003/) (2021)
- [Bayesian Model Selection via Mean-Field Variational Approximation](https://mlsanity.com/paper/2312.10607/) (2023)
- [On the Optimal Regret of Locally Private Linear Contextual Bandit](https://mlsanity.com/paper/2404.09413/) (2024)
- [Beyond Black Box Densities: Parameter Learning for the Deviated Components](https://mlsanity.com/paper/2202.02651/) (2022)
- [Fast Model Selection and Stable Optimization for Softmax-Gated Multinomial-Logistic Mixture of Experts Models](https://mlsanity.com/paper/2602.07997/) (2026)
- [Neural Kolmogorov Equations: Parallelizable Learning of Stochastic Dynamics under General Noise](https://mlsanity.com/paper/2607.19173/) (2026)

## Abstract (arXiv)

Singularities of a statistical model are the elements of the model's parameter space which make the corresponding Fisher information matrix degenerate. These are the points for which estimation techniques such as the maximum likelihood estimator and standard Bayesian procedures do not admit the root-n parametric rate of convergence. We propose a general framework for the identification of singularity structures of the parameter space of finite mixtures, and study the impacts of the singularity structures on minimax lower bounds and rates of convergence for the maximum likelihood estimator over a compact parameter space. Our study makes explicit the deep links between model singularities, parameter estimation convergence rates and minimax lower bounds, and the algebraic geometry of the parameter space for mixtures of continuous distributions. The theory is applied to establish concrete convergence rates of parameter estimation for finite mixture of skew-normal distributions. This rich and increasingly popular mixture model is shown to exhibit a remarkably complex range of asymptotic behaviors which have not been hitherto reported in the literature.