Quasi-likelihood estimation for GLM with random scales

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 typically used software tools such as weighted regression can compute 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.