The Burr 12 Distribution Family and the Maximum Entropy Principle: Power-Law Phenomena are not necessarily Nonextensive

TL;DR

This paper demonstrates how the Burr 12 distribution family can be derived from the maximum entropy principle, providing a stochastic interpretation and showing that power-law phenomena do not necessarily imply nonextensive entropy.

Contribution

It introduces a derivation of the Burr 12 distribution using entropy maximization and offers a new stochastic interpretation related to scale-dependent exponents.

Findings

Burr 12 distribution derived from maximum entropy principle

Power-law phenomena can be explained without nonextensive entropy

Distribution extensions include Pareto and loglogistic approximations

Abstract

In this paper we recall for physicists how it is possible, using the principle of maximization of the Boltzmann-Shannon entropy, to derive the Burr-Bingh-Maddala (burr12) double power law probability distribution function and its approximations (Pareto, loglogistic ..) and extension first used in econometrics. this is possible using a deformation of the power function, as this has been done in complex systems for the exponential function. We give to that distribution a deep stochastic interpretation using the theory of Weron et al. applied to thermodynamics the entropy nonextensivity can be accounted for by assuming that the asymptotic exponents are scale dependent. Therefore functions which describe phenomena presenting power-law asymptotic behaviour can be obtained without introducing exotic forms of the entropy.