Maximum Likelihood Estimation for Multinomial‐Poisson Models: A Generalization of Birch's Numerical Invariance Results

Abstract. This study gives a generalization of Birch's log‐linear model numerical invariance result. The generalization is given in the form of a sufficient condition for numerical invariance that is simple to verify in practice and is applicable for a much broader class of models than log‐linear models. Unlike Birch's log‐linear result, the generalization herein does not rely on any relationship between sufficient statistics and maximum likelihood estimates. Indeed the generalization does not rely on the existence of a reduced set of sufficient statistics. Instead, the concept of homogeneity takes centre stage. Several examples illustrate the utility of non‐log‐linear models, the invariance (and non‐invariance) of fitted values, and the invariance (and non‐invariance) of certain approximating distributions.