Maximum Likelihood Estimation in Log-Linear Models: Theory and Algorithms

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 non-existent MLE. Our conditions focus on the role of sampling zeros in the observed table. We situate our results within the general framework of extended exponential families and we rely in a fundamental way on key 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, especially as encapsulated in most standard computing packages, which appear to be oblivious to the problem of nonexistence of the MLE.