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.

arXiv commentv2: corrected equation (A5)

References in corpus (6)

• Anomalous diffusion and Tsallis statistics in an optical lattice
• The q-exponential family in statistical physics
• Generalised exponential families and associated entropy functions
• On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics
• q-Gaussians in the porous-medium equation: stability and time evolution
• Naudts-like duality and the extreme Fisher information principle

Cited by in corpus (5)

• Some properties of generalized Fisher information in the context of nonextensive thermostatistics
• On Generalized Stam Inequalities and Fisher-Rényi Complexity Measures
• One-parameter Fisher-Rényi complexity: Notion and hydrogenic applications
• On multidimensional generalized Cramér-Rao inequalities, uncertainty relations and characterizations of generalized $q$-Gaussian distributions
• Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions