On generalized Cram\'er-Rao inequalities, generalized Fisher informations and characterizations of generalized q-Gaussian distributions

TL;DR

This paper extends Cramér-Rao inequalities and Fisher information to nonextensive statistics, characterizing generalized q-Gaussian distributions and proposing new estimation methods based on these generalized concepts.

Contribution

It introduces generalized Cramér-Rao inequalities and Fisher information, characterizes q-Gaussian distributions, and suggests novel estimation techniques within nonextensive statistics.

Findings

Derived new extended Cramér-Rao inequalities involving q-moments.

Identified Cramér-Rao bounds saturated by q-Gaussian distributions.

Proposed generalized Fisher information and new estimation methods.

Abstract

This paper deals with Cram\'er-Rao inequalities in the context of nonextensive statistics and in estimation theory. It gives characterizations of generalized q-Gaussian distributions, and introduces generalized versions of Fisher information. The contributions of this paper are (i) the derivation of new extended Cram\'er-Rao inequalities for the estimation of a parameter, involving general q-moments of the estimation error, (ii) the derivation of Cram\'er-Rao inequalities saturated by generalized q-Gaussian distributions, (iii) the definition of generalized Fisher informations, (iv) the identification and interpretation of some prior results, and finally, (v) the suggestion of new estimation methods.