  When in a full exponential family the maximum likelihood estimate (MLE) does
not 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 on
repeated linear programming using the R contributed package rcdd and illustrate
it with two generalized linear model examples. When the MLE for the null
hypothesis lies in the completion, likelihood ratio tests of model comparison
are almost unchanged from the usual case. Only the degrees of freedom need to
be adjusted. When the MLE lies in the completion, confidence intervals are
changed much more from the usual case. The MLE of the natural parameter can be
thought of as having gone to infinity in a certain direction, which we call a
generic direction of recession. We propose a new one-sided confidence interval
which says how close to infinity the natural parameter may be. This maps to
one-sided confidence intervals for mean values showing how close to the
boundary of their support they may be.
