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Generalized maximum likelihood estimates for exponential families
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  • Published: 11 August 2007

Generalized maximum likelihood estimates for exponential families

  • Imre Csiszár1 &
  • František Matúš2 

Probability Theory and Related Fields volume 141, pages 213–246 (2008)Cite this article

  • 251 Accesses

  • 20 Citations

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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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References

  1. Barndorff-Nielsen O. (1978). Information and Exponential Families in Statistical Theory. Wiley, New York

    MATH  Google Scholar 

  2. Brown, L.D.: Fundamentals of Statistical Exponential Families. Inst. of Math. Statist. Lecture Notes–Monograph Series, Vol. 9 (1986)

  3. Chentsov, N.N.: Statistical Decision Rules and Optimal Inference. Translations of Mathematical Monographs, Amer. Math. Soc., Providence–Rhode Island, 1982 (Russian original: Nauka, Moscow, 1972)

  4. Csiszár I. and Matúš F. (2001). Convex cores of measures on \(\mathbb R^d\). Studia Sci. Math. Hungar. 38: 177–190

    MATH  MathSciNet  Google Scholar 

  5. Csiszár, I., Matúš, F.: Information closure of exponential families and generalized maximum likelihood estimates. In: Proc. 2002 IEEE Int. Symp. Inform. Theory, p. 434 (2002)

  6. Csiszár I. and Matúš F. (2003). Information projections revisited. IEEE Trans. Inform. Theory 49: 1474–1490

    Article  MATH  MathSciNet  Google Scholar 

  7. Csiszár I. and Matúš F. (2004). On information closures of exponential families: a counterexample. IEEE Trans. Inform. Theory 50: 922–924

    Article  MathSciNet  Google Scholar 

  8. Csiszár I. and Matúš F. (2005). Closures of exponential families. Ann. Probab. 33: 582–600

    Article  MATH  MathSciNet  Google Scholar 

  9. Csiszár, I., Matúš, F.: Generalized maximum likelihood estimates for infinite dimensional exponential families. In: Proceedings Prague Stochastics 2006. Prague, Czech Republic, pp. 288–297 (2006)

  10. Eriksson N., Fienberg S.E., Rinaldo A. and Sullivant S. (2006). Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models. J. Symbol. Comput. 41: 222–233

    Article  MATH  MathSciNet  Google Scholar 

  11. Letac, G.: Lectures on Natural Exponential Families and their Variance Functions. Monografias de Matemática 50. Instituto de Matemática Pura e Aplicada, Rio de Janeiro (1992)

  12. Lauritzen S.L. (1996). Graphical Models. Clarendon, Oxford

    Google Scholar 

  13. Rinaldo, A.: On maximum likelihood estimation in log-linear models. Technical Report 833, Department of Statistics, Carnegie Mellon University (2006)

  14. Rinaldo, A.: Computing maximum likelihood estimates in log-linear models. Technical Report 835. Department of Statistics, Carnegie Mellon University (2006)

  15. Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1364, Budapest, P.O. Box 127, Hungary

    Imre Csiszár

  2. Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 182 08, Prague, Czech Republic

    František Matúš

Authors
  1. Imre Csiszár
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  2. František Matúš
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Corresponding author

Correspondence to František Matúš.

Additional information

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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Cite this article

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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  • Received: 21 July 2006

  • Revised: 05 April 2007

  • Published: 11 August 2007

  • Issue Date: May 2008

  • DOI: https://doi.org/10.1007/s00440-007-0084-z

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Mathematics Subject Classification (2000)

  • Primary: 60A10
  • Secondary: 62H12
  • 62B10

Keywords

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