Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in Generalized Linear Models

Abstract

Generalized linear models are used for regression analysis in a number of cases, including categorical responses, where the classical assumptions are violated. The statistical analysis of such models is based on the asymptotic properties of the maximum likelihood estimator. We present mild general conditions which, respectively, assure weak or strong consistency or asymptotic normality. Most of the previous work has been concerned with natural link functions. In this case our normality condition, though obtained by a different approach, is closely related to a condition of Haberman (1977a). Examples show how the general conditions reduce to weak requirements for special exponential families. Further, for regressors with a compact range, sufficient conditions are given which do not involve the unknown parameter, and are therefore easy to check in practice. Responses with a bounded range, e.g. categorical responses, and stochastic regressors also are treated.