The subset argument and consistency of MLE in GLMM: Answer to an open problem and beyond

TL;DR

This paper addresses an open problem about the consistency of MLEs in GLMMs with crossed random effects, introducing a novel approach that extends to general models and demonstrates the technique with an example.

Contribution

It provides a new nonstandard method for proving MLE consistency in dependent data scenarios, solving an open problem and broadening applicability to general GLMMs.

Findings

Resolved an open problem on MLE consistency in GLMMs with crossed effects.

Developed a novel technique for dependent observations.

Extended results to general GLMMs with illustrative example.

Abstract

We give answer to an open problem regarding consistency of the maximum likelihood estimators (MLEs) in generalized linear mixed models (GLMMs) involving crossed random effects. The solution to the open problem introduces an interesting, nonstandard approach to proving consistency of the MLEs in cases of dependent observations. Using the new technique, we extend the results to MLEs under a general GLMM. An example is used to further illustrate the technique.