On consistency of the MLE under finite mixtures of location-scale distributions with a structural parameter

TL;DR

This paper proves the strong consistency of maximum likelihood estimators for finite mixtures of location-scale distributions, including various common densities, without requiring compact parameter spaces, and extends results to multivariate elliptical distributions.

Contribution

It provides a rigorous proof of MLE consistency for finite mixture models with a structural parameter, applicable to multiple distribution types and multivariate cases.

Findings

MLEs are strongly consistent for various location-scale mixtures.

Consistency holds without compactness assumptions on parameters.

Results extend to multivariate elliptical distributions.

Abstract

We provide a general and rigorous proof for the strong consistency of maximum likelihood estimators of the cumulative distribution function of the mixing distribution and structural parameter under finite mixtures of location-scale distributions with a structural parameter. The consistency results do not require the parameter space of location and scale to be compact. We illustrate the results by applying them to finite mixtures of location-scale distributions with the component density function being one of the commonly used density functions: normal, logistic, extreme-value, or $t$. An extension of the strong consistency results to finite mixtures of multivariate elliptical distributions is also discussed.