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Escort distributions minimizing the Kullback–Leibler divergence for a large deviations principle and tests of entropy level

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  • Valérie Girardin
  • Philippe Regnault

Abstract

Kullback–Leibler divergence is minimized among finite distributions with finite state spaces under various constraints of Shannon entropy. Minimization is closely linked to escort distributions whose main properties related to entropy are proven. This allows a large deviations principle to be stated for the sequence of plug-in empirical estimators of Shannon entropy of any finite distributions. Since no closed-form expression of the rate function can be obtained, an explicit approximating function is constructed. This approximation is accurate enough to provide good results in all applications. Tests of entropy level, using both the large deviations principle and the minimization results, are constructed and shown to have a good behavior in terms of errors. Copyright The Institute of Statistical Mathematics, Tokyo 2016

Suggested Citation

  • Valérie Girardin & Philippe Regnault, 2016. "Escort distributions minimizing the Kullback–Leibler divergence for a large deviations principle and tests of entropy level," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(2), pages 439-468, April.
    • Handle: RePEc:spr:aistmt:v:68:y:2016:i:2:p:439-468
    DOI: 10.1007/s10463-014-0501-x
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    Cited by:

    1. Asok K. Nanda & Shovan Chowdhury, 2021. "Shannon's Entropy and Its Generalisations Towards Statistical Inference in Last Seven Decades," International Statistical Review, International Statistical Institute, vol. 89(1), pages 167-185, April.
    2. Valérie Girardin & Loick Lhote & Philippe Regnault, 2019. "Different Closed-Form Expressions for Generalized Entropy Rates of Markov Chains," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1431-1452, December.

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