Computationally efficient likelihood inference in exponential families when the maximum likelihood estimator does not exist
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In a regular full exponential family, the maximum likelihood estimator (MLE) need not exist, in the traditional sense, but the MLE may exist in the Barndorff-Nielsen completion of the family. Existing algorithms for finding the MLE in the Barndorff-Nielsen 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 Barndorff-Nielsen completion based on approximate null eigenvectors of the Fisher information matrix. Convergence of Fisher information follows from cumulant generating function convergence, conditions for which are given.
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