  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.
