On Statistical Properties of Jizba-Arimitsu Hybrid Entropy

TL;DR

This paper explores the statistical properties of Jizba-Arimitsu hybrid entropy, highlighting its unique features, differences from Rnyi and Tsallis entropies, and introducing an average hybrid entropy concept.

Contribution

It provides a comprehensive analysis of hybrid entropy's properties, its continuous form, divergence, and introduces an average hybrid entropy measure.

Findings

Hybrid entropy has distinct properties from Rnyi and Tsallis entropies.

Derived hybrid entropy for well-known distributions.

Introduced average hybrid entropy as a new measure.

Abstract

Jizba-Arimitsu entropy (also called hybrid entropy) combines axiomatics of R\'enyi and Tsallis entropy. It has many common properties with them, on the other hand, some aspects as e.g., MaxEnt distributions, are completely different from the former two entropies. In this paper, we demonstrate the statistical properties of hybrid entropy, including the definition of hybrid entropy for continuous distributions, its relation to discrete entropy and calculation of hybrid entropy for some well-known distributions. Additionally, definition of hybrid divergence and its connection to Fisher metric is also discussed. Interestingly, the main properties of continuous hybrid entropy and hybrid divergence are completely different from measures based on R\'enyi and Tsallis entropy. This motivates us to introduce average hybrid entropy, which can be understood as an average between Tsallis and R\'enyi entropy.