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

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.