Generalized entropy optimized by an arbitrary distribution

TL;DR

This paper develops a generalized entropy framework optimized for arbitrary distributions with finite linear expectations, unifying various natural distributions and extending to those with divergent moments.

Contribution

It introduces a method to construct generalized entropy for any distribution with finite expectation, including those with divergent moments like Levy stable distributions.

Findings

Derived explicit entropy for stretched exponential distribution

Unified description of diverse natural distributions

Extended entropy definition to include divergent moments

Abstract

We construct the generalized entropy optimized by a given arbitrary statistical distribution with a finite linear expectation value of a random quantity of interest. This offers, via the maximum entropy principle, a unified basis for a great variety of distributions observed in nature, which can hardly be described by the conventional methods. As a simple example, we explicitly derive the entropy associated with the stretched exponential distribution. To include the distributions with the divergent moments (e.g., the Levy stable distributions), it is necessary to modify the definition of the expectation value.