Maximum Entropy Framework for a Universal Rank Order distribution with Socio-economic Applications

TL;DR

This paper derives the maximum entropy properties of the discrete generalized beta distribution, explaining its success in modeling rank-size data across various fields and exploring its socio-economic applications.

Contribution

It provides the first theoretical derivation of the distribution as a maximum entropy model under a utility constraint, linking it to socio-economic modeling.

Findings

The distribution fits empirical data well across contexts.

Estimated parameters characterize socio-economic factors over time.

Entropy measures quantify distributional changes.

Abstract

In this paper we derive the maximum entropy characteristics of a particular rank order distribution, namely the discrete generalized beta distribution, which has recently been observed to be extremely useful in modelling many several rank-size distributions from different context in Arts and Sciences, as a two-parameter generalization of Zipf's law. Although it has been seen to provide excellent fits for several real world empirical datasets, the underlying theory responsible for the success of this particular rank order distribution is not explored properly. Here we, for the first time, provide its generating process which describes it as a natural maximum entropy distribution under an appropriate bivariate utility constraint. Further, considering the similarity of the proposed utility function with the usual logarithmic utility function from economic literature, we have also explored its acceptability in universal modeling of different types of socio-economic factors within a country as well as across the countries. The values of distributional parameters estimated through a rigorous statistical estimation method, along with the $entropy$ values, are used to characterize the distributions of all these socio-economic factors over the years.