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Generalized maximum likelihood estimates for exponential families. (English) Zbl 1133.62039
Summary: 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 the 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.

62H12 Estimation in multivariate analysis
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60A10 Probabilistic measure theory
Full Text: DOI
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