Generalized maximum likelihood estimates for exponential families

For a standard full exponential family on Ropf^d, or its canonically convex subfamily, the generalized maximum likelihood estimator is an extension of the mapping that assigns to the mean alpha isin Ropf^d of a sample for which a maximizer v* of the corresponding likelihood function exists, the member of the family parameterized by v*. This extension assigns to each alpha; isin Ropf^d with the likelihood function bounded above, a member of the closure of the family in variation distance. Its detailed description, complete characterization of domain and range, and additional results are presented, in a general setting. In addition to basic convex analysis tools, the authors´ prior results on convex cores of measures and closures of exponential families are used