Dispersion Parameter Extension of Precise Generalized Linear Mixed Model Asymptotics

TL;DR

This paper extends the asymptotic theory for generalized linear mixed models to include the dispersion parameter, showing that all estimators are asymptotically normal and independent, simplifying inference procedures.

Contribution

It introduces an extension of asymptotic normality results to the dispersion parameter in generalized linear mixed models, enabling simpler likelihood-based inference.

Findings

Estimators of all parameters are asymptotically normal and mutually independent.

Dispersion parameter estimator has a simple asymptotic distribution.

Facilitates straightforward likelihood-based inference for GLMMs.

Abstract

We extend a recently established asymptotic normality theorem for generalized linear mixed models to include the dispersion parameter. The new results show that the maximum likelihood estimators of all model parameters have asymptotically normal distributions with asymptotic mutual independence between fixed effects, random effects covariance and dispersion parameters. The dispersion parameter maximum likelihood estimator has a particularly simple asymptotic distribution which enables straightforward valid likelihood-based inference.