Characterization of Generalized Alpha-Beta Divergence and 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.