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When in a full exponential family the maximum likelihood estimate (MLE) doesnot exist, the MLE may exist in the Barndorff-Nielsen completion of the family.We propose a practical algorithm for finding the MLE in the completion based onrepeated linear programming using the R contributed package rcdd and illustrateit with two generalized linear model examples. When the MLE for the nullhypothesis lies in the completion, likelihood ratio tests of model comparisonare almost unchanged from the usual case. Only the degrees of freedom need tobe adjusted. When the MLE lies in the completion, confidence intervals arechanged much more from the usual case. The MLE of the natural parameter can bethought of as having gone to infinity in a certain direction, which we call ageneric direction of recession. We propose a new one-sided confidence intervalwhich says how close to infinity the natural parameter may be. This maps toone-sided confidence intervals for mean values showing how close to theboundary of their support they may be.

Likelihood Inference in Exponential Families and Directions of Recession