The compactification of generalized linear models

Abstract

The main purpose of this paper is to unify and extend the existing theory of 'estimated zeroes' in log-linear and logit models. To this end it is shown, that every GLM can be embedded in a larger model with a compact parameter space and a continuous likelihood (a 'CGLM'). Clearly in a CGLM the MLE always exists, solving a major data-analysis problem. In the mean value parametrization the construction of the CGLM is remarkably simple; in theß-parametrization it is more complex. Estimated expected values are always finite, but the MLE need not correspond with a finite β̂, as is well known for estimated zeroes in log-linear models. The boundary distributions of CGLMs are classified in four categories: 'inadmissable', 'degenerate', 'Chentsov', and 'constrained'. For a large class of GLMs, including all GLMs with canonical link functions and probit models, the MLE in the corresponding CGLM exists and is unique. This seems to be new for log-linear models. We give equivalent algebraic and geometric conditions (in the vein of Haberman (1974, 1977) and Albert & Anderson (1984) respectively), necessary for the existence of the MLE in the GLM (corresponding to a finite β̂). For a large class of GLMs these conditions are also sufficient. This too seems new for log-linear models.