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

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.