Strong consistency of the maximum likelihood estimator for finite mixtures of location-scale distributions when penalty is imposed on the ratios of the scale parameters

TL;DR

This paper proves the strong consistency of penalized maximum likelihood estimators in finite location-scale mixture models, addressing the unbounded likelihood issue by imposing penalties on scale ratios or parameters.

Contribution

It introduces a novel penalized likelihood approach with penalties on scale ratios, ensuring the existence and strong consistency of the MLE in finite mixture models.

Findings

Penalized MLE is strongly consistent under certain conditions.

Penalties on scale ratios prevent unbounded likelihood.

Consistency also holds when penalizing scale parameters directly.

Abstract

In finite mixtures of location-scale distributions, if there is no constraint or penalty on the parameters, then the maximum likelihood estimator does not exist because the likelihood is unbounded. To avoid this problem, we consider a penalized likelihood, where the penalty is a function of the minimum of the ratios of the scale parameters and the sample size. It is shown that the penalized maximum likelihood estimator is strongly consistent. We also analyze the consistency of a penalized maximum likelihood estimator where the penalty is imposed on the scale parameters themselves.