Characterization of Generalized Alpha-Beta Divergence and Associated Entropy Measures

TL;DR

This paper introduces a new generalized alpha-beta divergence family that encompasses existing divergence measures, providing a unified framework with novel properties and associated entropy measures for statistical inference.

Contribution

It proposes a comprehensive generalized divergence measure, establishes conditions for its validity, and explores its properties and related entropy measures, unifying many existing divergence families.

Findings

Introduces generalized alpha-beta divergence as a superfamily of existing divergences.

Derives conditions for the validity of the new divergence measure.

Explores properties like duality, inversion, and semi-continuity, and discusses associated entropy measures.

Abstract

Minimum divergence estimators provide a natural choice of estimators in a statistical inference problem. Different properties of various families of these divergence measures such as Hellinger distance, power divergence, density power divergence, logarithmic density power divergence, etc. have been established in literature. In this work, we propose a new class of divergence measures called "generalized alpha-beta divergence", which is a superfamily of these popular divergence families. We provide the necessary and sufficient conditions for the validity of the proposed generalized divergence measure, which allows us to construct novel families of divergence and associated entropy measures. We also show various characterizing properties like duality, inversion, semi-continuity, etc., from which, many existing results follow as special cases. We also discuss about the entropy measure derived from this general family of divergence and its properties.