Abstract
For a standard full exponential family on \(\mathbb R^d\) , or its canonically convex subfamily, the generalized maximum likelihood estimator is an extension of the mapping that assigns to the mean \(a\in\mathbb R^d\) of a sample for which a maximizer \(\vartheta^*\) of a corresponding likelihood function exists, the member of the family parameterized by \(\vartheta^*\) . This extension assigns to each \(a\in\mathbb R^d\) 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, 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.
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This paper is dedicated to the memory of Albert Perez (1920–2003).
This work was supported by the Hungarian National Foundation for Scientific Research under Grant T046376 and by Grant Agency of Academy of Sciences of the Czech Republic under Grant IAA 100750603.
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Csiszár, I., Matúš, F. Generalized maximum likelihood estimates for exponential families. Probab. Theory Relat. Fields 141, 213–246 (2008). https://doi.org/10.1007/s00440-007-0084-z
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DOI: https://doi.org/10.1007/s00440-007-0084-z