Bounds on f-Divergences between Distributions within Generalized Quasi-$\varepsilon$-Neighborhood

TL;DR

This paper develops computable bounds on f-divergences within a generalized neighborhood framework, unifying local distributional proximity, extending divergence classification, and improving reverse Pinsker's inequalities for better statistical testing.

Contribution

It introduces a unified framework for bounding f-divergences, generalizes divergence classification, and provides tighter inequalities for statistical analysis.

Findings

Unified characterization of local distributional proximity.

Taylor-based inequalities for f-divergence classification.

Tighter reverse Pinsker's inequalities for asymptotic analysis.

Abstract

This work establishes computable bounds between f-divergences for probability measures within a generalized quasi-$\varepsilon_{(M,m)}$-neighborhood framework. We make the following key contributions. (1) a unified characterization of local distributional proximity beyond structural constraints is provided, which encompasses discrete/continuous cases through parametric flexibility. (2) First-order differentiable $f$-divergence classification with Taylor-based inequalities is established, which generalizes $\chi^2$-divergence results to broader function classes. (3) We provide tighter reverse Pinsker's inequalities than existing ones, bridging asymptotic analysis and computable bounds. The proposed framework demonstrates particular efficacy in goodness-of-fit test asymptotics while maintaining computational tractability.