Maximum likelihood estimation in log-linear models

TL;DR

This paper investigates maximum likelihood estimation in log-linear models with a focus on the impact of sampling zeros, providing conditions for MLE existence, analyzing estimability issues, and proposing improved algorithms within the extended exponential family framework.

Contribution

It offers new necessary and sufficient conditions for MLE existence in log-linear models under sampling zeros and introduces improved algorithms for extended maximum likelihood estimation.

Findings

Conditions for MLE existence based on sampling zeros

Analysis of estimability of parameters when MLE does not exist

Enhanced algorithms for extended maximum likelihood estimation

Abstract

We study maximum likelihood estimation in log-linear models under conditional Poisson sampling schemes. We derive necessary and sufficient conditions for existence of the maximum likelihood estimator (MLE) of the model parameters and investigate estimability of the natural and mean-value parameters under a nonexistent MLE. Our conditions focus on the role of sampling zeros in the observed table. We situate our results within the framework of extended exponential families, and we exploit the geometric properties of log-linear models. We propose algorithms for extended maximum likelihood estimation that improve and correct the existing algorithms for log-linear model analysis.