Context-sensitive hypothesis-testing and exponential families

TL;DR

This paper introduces concepts in weighted statistical inference, extending classical bounds and divergences to improve hypothesis testing within exponential families, with applications in various scientific fields.

Contribution

It generalizes key statistical inequalities and divergences to weighted contexts, enhancing hypothesis testing methods for exponential families.

Findings

Generalized Stein-Sanov theorem for weighted data

Extended divergence measures like Kullback-Leibler and Chernoff

New bounds and asymptotics for weighted inference

Abstract

We propose a number of concepts and properties related to `weighted' statistical inference where the observed data are classified in accordance with a `value' of a sample string. The motivation comes from the concepts of weighted information and weighted entropy that proved useful in industrial/microeconomic and medical statistics. We focus on applications relevant in hypothesis testing and an analysis of exponential families. Several notions, bounds and asymptotics are established, which generalize their counterparts well-known in standard statistical research. It includes Stein-Sanov theorem, Pinsker's, Bretangnole-Huber and van Trees inequalities and Kullback--Leibler, Bhattacharya, Bregman, Burbea-Rao, Chernoff, Renyi and Tsallis divergences.