Generalized maximum likelihood estimates for exponential families

For a standard full exponential family on , or its canonically convex subfamily, the generalized maximum likelihood estimator is an extension of the mapping that assigns to the mean of a sample for which a maximizer of a corresponding likelihood function exists, the member of the family parameterized by . This extension assigns to each 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, not imposing any regularity assumptions. In addition to basic convex analysis tools, the authors’ prior results on convex cores of measures and closures of exponential families are used.