Computationally efficient likelihood inference in exponential families when the maximum likelihood estimator does not exist

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