# Hypothesis Testing via Euclidean Separation

## TL;DR

This paper introduces a convex optimization-based approach for hypothesis testing in heavy-tailed distributions, utilizing Euclidean separation of convex sets to develop quasi-optimal tests and applying them to sequential detection problems.

## Contribution

It extends hypothesis testing methods to heavy-tailed distributions using Euclidean separation, providing a new framework for constructing quasi-optimal tests and applying them to dynamic system change detection.

## Key findings

- Developed a convex optimization-based hypothesis testing framework for heavy-tailed distributions.
- Constructed quasi-optimal tests for families majorated by sub-spherical symmetric distributions.
- Applied the methodology to sequential detection of system input changes.

## Abstract

We discuss an "operational" approach to testing convex composite hypotheses when the underlying distributions are heavy-tailed. It relies upon Euclidean separation of convex sets and can be seen as an extension of the approach to testing by convex optimization developed in [8, 12]. In particular, we show how one can construct quasi-optimal testing procedures for families of distributions which are majorated, in a certain precise sense, by a sub-spherical symmetric one and study the relationship between tests based on Euclidean separation and "potential-based tests." We apply the promoted methodology in the problem of sequential detection and illustrate its practical implementation in an application to sequential detection of changes in the input of a dynamic system. [8] Goldenshluger, Alexander and Juditsky, Anatoli and Nemirovski, Arkadi, Hypothesis testing by convex optimization, Electronic Journal of Statistics,9 (2):1645-1712, 2015. [12] Juditsky, Anatoli and Nemirovski, Arkadi, Hypothesis testing via affine detectors, Electronic Journal of Statistics, 10:2204--2242, 2016.