Strong consistency of MLE for finite mixtures of location-scale distributions when the ratios of the scale parameters are exponentially small

TL;DR

This paper proves that the maximum likelihood estimator for finite mixtures of location-scale distributions is strongly consistent when the ratios of scale parameters are restricted to be exponentially small in the sample size.

Contribution

It establishes the strong consistency of MLE under a specific exponential restriction on scale parameter ratios, filling a gap in theoretical understanding.

Findings

MLE is strongly consistent under exponential scale ratio restrictions

Existence of MLE is guaranteed with these constraints

Provides theoretical foundation for finite mixture models with restricted parameters

Abstract

In finite mixtures of location-scale distributions, if there is no constraint on the parameters then the maximum likelihood estimate does not exist. But when the ratios of the scale parameters are restricted appropriately, the maximum likelihood estimate exists. We prove that the maximum likelihood estimator (MLE) is strongly consistent, if the ratios of the scale parameters are restricted from below by $\exp(-n^{d}), 0 < d < 1 $, where $n$ is the sample size.