  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.
