  We consider a linear mixed-effects model with a clustered structure, where
the parameters are estimated using maximum likelihood (ML) based on possibly
unbalanced data. Inference with this model is typically done based on
asymptotic theory, assuming that the number of clusters tends to infinity with
the sample size. However, when the number of clusters is fixed, classical
asymptotic theory developed under a divergent number of clusters is no longer
valid and can lead to erroneous conclusions. In this paper, we establish the
asymptotic properties of the ML estimators of random-effects parameters under a
general setting, which can be applied to conduct valid statistical inference
with fixed numbers of clusters. Our asymptotic theorems allow both fixed
effects and random effects to be misspecified, and the dimensions of both
effects to go to infinity with the sample size.
