Generalized maximum likelihood estimates for exponential families

Imre Csiszár, František Matúš

Research output: Contribution to journalArticle

20 Citations (Scopus)


For a standard full exponential family on ℝd, or its canonically convex subfamily, the generalized maximum likelihood estimator is an extension of the mapping that assigns to the mean a ∞ ℝd of a sample for which a maximizer v* of a corresponding likelihood function exists, the member of the family parameterized by v*. This extension assigns to each a ∞ ℝd 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.

Original languageEnglish
Pages (from-to)213-246
Number of pages34
JournalProbability Theory and Related Fields
Issue number1-2
Publication statusPublished - May 1 2008



  • Accessible face
  • Convex core
  • Exponential family
  • Kullback-Leibler information divergence
  • Log-convexity
  • Maximum likelihood
  • Partial mean
  • Variation distance closure

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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