The compactification of generalized linear models

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 generalized linear model (GLM) can be embedded in a larger model with a compact parameter space and a continuous likelihood (a 'CGLM'). Clearly in a CGLM the maximum likelihood estimate (MLE) always exists, easing a major data analysis problem. In the mean-value parametrization, the construction of the CGLM is remarkably simple; except in a rather pathological and rare case, the estimated expected values are always finite., In the β-parametrization however, the compactification is more complex; the MLE need not correspond with a finite β, as is well known for estimated zeros in log-linear models. The boundary distributions of CGLMs are classified in four categories: 'Inadmissible', '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. Even stronger, the likelihood has no other local maxima. We give equivalent algebraic and geometric conditions (in the vein of Haberman (1974, 1977) and Albert and Anderson (1984) respectively), necessary for the existence of the MLE in the GLM corresponding to a finite β. For a large class of GLMs these conditions are also sufficient. Even for log-linear models this seams to be a new result.