Bounds smaller than the Fisher information for generalized linear models

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.