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

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.