  In this paper, we propose a parameter space augmentation approach that is
based on "intentionally" introducing a pseudo-nuisance parameter into
generalized linear models for the purpose of variance reduction. We first
consider the parameter whose norm is equal to one. By introducing a
pseudo-nuisance parameter into models to be estimated, an extra estimation is
asymptotically normal and is, more importantly, non-positively correlated to
the estimation that asymptotically achieves the Fisher/quasi Fisher
information. As such, the resulting estimation is asymptotically with smaller
variance-covariance matrices than the Fisher/quasi Fisher information. For
general cases where the norm of the parameter is not necessarily equal to one,
two-stage quasi-likelihood procedures separately estimating the scalar and
direction of the parameter are proposed. The traces of the limiting
variance-covariance matrices are in general smaller than or equal to that of
the Fisher/quasi-Fisher information. We also discuss the pros and cons of the
new methodology, and possible extensions. As this methodology of parameter
space augmentation is general, and then may be readily extended to handle, say,
cluster data and correlated data, and other models.
