Geometry of escort distributions

TL;DR

This paper explores the geometric structure of escort distributions using information geometry, revealing their Fisher metric, connection to multifractal fluctuations, and implications for statistical estimation in nonextensive mechanics.

Contribution

It introduces a geometric framework for escort distributions based on Kullback-Leibler divergence and Fisher metric, linking them to multifractal fluctuations and estimation limits.

Findings

Fisher metric expressed via generalized bit-variance

Cramer-Rao inequality sets fundamental estimation limits

Using original distributions instead of escort distributions can lead to inaccuracies

Abstract

Given an original distribution, its statistical and probabilistic attributs may be scanned by the associated escort distribution introduced by Beck and Schlogl and employed in the formulation of nonextensive statistical mechanics. Here, the geometric structure of the one-parameter family of the escort distributions is studied based on the Kullback-Leibler divergence and the relevant Fisher metric. It is shown that the Fisher metric is given in terms of the generalized bit-variance, which measures fluctuations of the crowding index of a multifractal. The Cramer-Rao inequality leads to the fundamental limit for precision of statistical estimate of the order of the escort distribution. It is also quantitatively discussed how inappropriate it is to use the original distribution instead of the escort distribution for calculating the expectation values of physical quantities in nonextensive statistical mechanics.