A mixture of a normal distribution with random mean and variance -- Examples of inconsistency of maximum likelihood estimates

TL;DR

This paper investigates the estimation challenges in a normal mixture model with random mean and variance, highlighting issues of non-identifiability and inconsistency of maximum likelihood estimators, and conditions for identifiability.

Contribution

It provides theoretical insights into the identifiability and inconsistency of estimators in normal mixture models with random parameters, including conditions for identifiability.

Findings

Model is generally not identifiable.

MLE is inconsistent even with bounded shift and independent parameters.

Mixing distribution is identifiable with multiple observations per latent realization.

Abstract

We consider the estimation of the mixing distribution of a normal distribution where both the shift and scale are unobserved random variables. We argue that in general, the model is not identifiable. We give an elegant non-constructive proof that the model is identifiable if the shift parameter is bounded by a known value. However, we argue that the generalized maximum likelihood estimator is inconsistent even if the shift parameter is bounded and the shift and scale parameters are independent. The mixing distribution, however, is identifiable if we have more than one observations per any realization of the latent shift and scale.