On a (\beta,q)-generalized Fisher information and inequalities involving q-Gaussian distributions

TL;DR

This paper introduces a generalized Fisher information within nonextensive thermostatistics, showing it leads to generalized q-Gaussians, a new Cramér-Rao inequality, and extended Stam inequalities, enriching the understanding of information measures in this framework.

Contribution

It defines a new generalized Fisher information that aligns with nonextensive thermostatistics and demonstrates its role in deriving inequalities and characterizing q-Gaussian distributions.

Findings

Generalized Fisher information minimizes at q-Gaussians.

Derived a generalized Cramér-Rao inequality.

Established an extended Stam inequality.

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 R\'enyi 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 Cram\'er-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.