Model checking in loglinear models using φ-divergences and MLEs

Abstract

Consider the loglinear model for categorical data under the assumption of either Poisson, multinomial, or product-multinomial sampling. We are interested in testing between various hypotheses on the parameter space. In this paper, the usual likelihood ratio test, with maximum likelihood estimators for the unspecified parameters, is generalized to tests based on φ-divergences, still using maximum likelihood estimators. These tests yield the likelihood ratio test as a special case. Asymptotic distributions for the new test statistics are derived under both the null and the alternative hypotheses. Then it is shown how the φ-divergences can be used to test nested hypotheses, yielding a type of “analysis of divergence”.