MAXIMUM LIKELIHOOD ESTIMATION WITH BINARY-DATA REGRESSION MODELS: SMALL-SAMPLE AND LARGE-SAMPLE FEATURES

Roland C. Deutsch, John M. Grego, Brian Habing and Walter W. Piegorsch

Received July 2, 2009

Abstract

Many inferential procedures for generalized linear models (GLiMs) rely on the asymptotic normality of the maximum likelihood estimator (MLE). Fahrmeir and Kaufmann [5] present mild conditions under which the MLEs in GLiMs are asymptotically normal. Unfortunately, limited study has appeared for the special case of binomial response models beyond the familiar logit and probit links, with little results for more general links such as the complementary log-log link, and the less well-known complementary log link. We verify the asymptotic normality conditions of the MLEs for these models under the assumption of a fixed number of experimental groups and present a simple set of conditions for any twice-differentiable monotone link function. We also study the quality of the approximation for constructing asymptotic Wald confidence regions. Our results show that for small sample sizes with certain link functions the approximation can be problematic, especially for cases where the parameters are close to the boundary of the parameter space.

Keywords and phrases: binomial response models, maximum likelihood estimator, asymptotic normality, complementary log-log link, complementary log link.