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The three-parameter generalized extreme value distribution arises fromclassical univariate extreme value theory and is in common use for analyzingthe far tail of observed phenomena. Curiously, important asymptotic propertiesof likelihood-based estimation under this standard model have yet to beestablished. In this paper, we formally prove that the maximum likelihoodestimator is global and unique. An interesting secondary result entails theuniform consistency of a class of limit relations in a tight neighborhood ofthe shape parameter.

Uniqueness and global optimality of the maximum likelihood estimator for the generalized extreme value distribution