# On a (β,q)-generalized Fisher information and inequalities involving q-Gaussian distributions

Abstract : In the present paper, we would like to draw attention to a possible generalized Fisher information that fits well in the formalism of nonextensive thermostatistics. This generalized Fisher information is defined for densities on $\mathbb{R}^{n}.$ Just as the maximum Rényi or Tsallis entropy subject to an elliptic moment constraint is a generalized q-Gaussian, we show that the minimization of the generalized Fisher information also leads a generalized q-Gaussian. This yields a generalized Cramér-Rao inequality. In addition, we show that the generalized Fisher information naturally pops up in a simple inequality that links the generalized entropies, the generalized Fisher information and an elliptic moment. Finally, we give an extended Stam inequality. In this series of results, the extremal functions are the generalized q-Gaussians. Thus, these results complement the classical characterization of the generalized q-Gaussian and introduce a generalized Fisher information as a new information measure in nonextensive thermostatistics.
Type de document :
Article dans une revue
Journal of Mathematical Physics, American Institute of Physics (AIP), 2012, 53, pp.063303. 〈10.1063/1.4726197〉

Littérature citée [37 références]

https://hal.archives-ouvertes.fr/hal-00733758
Contributeur : Jean-François Bercher <>
Soumis le : mercredi 19 septembre 2012 - 14:27:06
Dernière modification le : mercredi 11 avril 2018 - 12:12:02
Document(s) archivé(s) le : vendredi 16 décembre 2016 - 15:17:46

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OnqGenFisher_arxiv2.pdf
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### Citation

Jean-François Bercher. On a (β,q)-generalized Fisher information and inequalities involving q-Gaussian distributions. Journal of Mathematical Physics, American Institute of Physics (AIP), 2012, 53, pp.063303. 〈10.1063/1.4726197〉. 〈hal-00733758〉

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