This paper uses random scales similar to random effects used
in the generalized linear mixed models to describe
"inter-location" population variation in variance components
for modeling complicated data obtained from applications such as
antenna manufacturing. Our distribution studies lead to
a complicated integrated extended quasi-likelihood (IEQL) for
parameter estimations and large sample inference derivations.
Laplace's expansion and several approximation methods
are employed to simplify the IEQL estimation procedures.
Asymptotic properties of the approximate IEQL estimates are
derived for general structures of the covariance matrix of random scales.
Focusing on a few special covariance structures in simpler forms,
the authors further simplify IEQL estimates such that the typically used software
tools such as weighted regression can perform the estimates easily.
Moreover, these special cases
allow us to derive interesting asymptotic results in much more
compact expressions. Finally, numerical simulation results show that IEQL estimates
perform very well in several special cases studied.