Some results on a $\chi$-divergence, an~extended~Fisher information and~generalized~Cram\'er-Rao inequalities

TL;DR

This paper introduces a modified chi-divergence and a generalized Fisher information, leading to extended Cramér-Rao inequalities and new characterizations of generalized q-Gaussians, with potential implications for uncertainty relations.

Contribution

It proposes a new chi-divergence, defines a generalized Fisher information, and extends classical inequalities and characterizations in statistical estimation theory.

Findings

Introduces a modified chi-divergence with new properties.

Develops generalized Cramér-Rao inequalities involving the new Fisher information.

Provides new characterizations of generalized q-Gaussians and potential uncertainty relations.

Abstract

We propose a modified $\chi^{\beta}$-divergence, give some of its properties, and show that this leads to the definition of a generalized Fisher information. We give generalized Cram\'er-Rao inequalities, involving this Fisher information, an extension of the Fisher information matrix, and arbitrary norms and power of the estimation error. In the case of a location parameter, we obtain new characterizations of the generalized $q$-Gaussians, for instance as the distribution with a given moment that minimizes the generalized Fisher information. Finally we indicate how the generalized Fisher information can lead to new uncertainty relations.