  In a regular full exponential family, the maximum likelihood estimator (MLE)
need not exist in the traditional sense. However, the MLE may exist in the
completion of the exponential family. Existing algorithms for finding the MLE
in the completion solve many linear programs; they are slow in small problems
and too slow for large problems. We provide new, fast, and scalable methodology
for finding the MLE in the completion of the exponential family. This
methodology is based on conventional maximum likelihood computations which come
close, in a sense, to finding the MLE in the completion of the exponential
family. These conventional computations construct a likelihood maximizing
sequence of canonical parameter values which goes uphill on the likelihood
function until they meet a convergence criteria. Nonexistence of the MLE in
this context results from a degeneracy of the canonical statistic of the
exponential family, the canonical statistic is on the boundary of its support.
There is a correspondance between this boundary and the null eigenvectors of
the Fisher information matrix. Convergence of Fisher information along a
likelihood maximizing sequence follows from cumulant generating function (CGF)
convergence along a likelihood maximizing sequence, conditions for which are
given. This allows for the construction of necessarily one-sided confidence
intervals for mean value parameters when the MLE exists in the completion. We
demonstrate our methodology on three examples in the main text and three
additional examples in the Appendix. We show that when the MLE exists in the
completion of the exponential family, our methodology provides statistical
inference that is much faster than existing techniques.
