Nested Inequalities Among Divergence Measures

TL;DR

This paper establishes a comprehensive inequality involving 11 divergence measures, creating a hierarchy of over 40 measures and introducing reverse inequalities, advancing understanding in information theory and statistics.

Contribution

It introduces a unifying inequality among multiple divergence measures and constructs a nested sequence of over 40 measures, including reverse inequalities, enhancing theoretical insights.

Findings

Established a single inequality involving 11 divergence measures.

Created a nested sequence of over 40 divergence measures.

Introduced the concept of reverse inequalities.

Abstract

In this paper we have considered a single inequality having 11 known divergence measures. This inequality include measures like: Jeffryes-Kullback-Leiber J-divergence, Jensen-Shannon divergence (Burbea-Rao, 1982), arithmetic-geometric mean divergence (Taneja, 1995), Hellinger discrimination, symmetric chi-square divergence, triangular discrimination, etc. All these measures are well-known in the literature on Information theory and Statistics. This sequence of 11 measures also include measures due to Kumar and Johnson (2005) and Jain and Srivastava (2007). Three measures arising due to some mean divergences also appears in this inequality. Based on non-negative differences arising due to this single inequality of 11 measures, we have put more than 40 divergence measures in nested or sequential form. Idea of reverse inequalities is also introduced.