On the existence of maximum likelihood estimates in logistic regression models

The problems of existence, uniqueness and location of maximum likelihood estimates in log linear models have received special attention in the literature (Haberman, 1974, Chapter 2; Wedderburn, 1976; Silvapulle, 1981). For multinomial logistic regression models, we prove existence theorems by considering the possible patterns of data points, which fall into three mutually exclusive and exhaustive categories: complete separation, quasicomplete separation and overlap. Our results suggest general rules for identifying infinite parameter estimates in log linear models for frequency tables.