Optimal e-values for testing the mean of a bounded random variable against a composite alternative

TL;DR

This paper derives uniquely optimal e-values for testing the mean of bounded variables, extending the application of (RE)GROW criteria beyond mutually absolutely continuous hypotheses, and identifies challenging alternatives for these tests.

Contribution

It introduces the first application of (RE)GROW criteria to bounded variables, explicitly characterizes difficult testing scenarios, and demonstrates the practical utility of REGROW over GROW.

Findings

Derived unique optimal e-values for bounded variables.

Explicitly characterized the most difficult testing alternatives.

Showed REGROW's effectiveness in practical examples.

Abstract

We derive the unique e-values with optimal (relative) growth rate in the worst case for testing the mean of a bounded random variable, hereby contributing with the first application beyond the assumption of mutually absolutely continuous hypotheses of the (RE)GROW quality criteria for e-values originally proposed by Gr\"unwald et al. (2024). For both criteria, we characterise explicitly the alternatives for which it is most difficult to test against, which also admit a meaningful interpretation. We give two important examples of interest where REGROW provides a powerful quality criterion to choose optimal e-variables whereas GROW leads to trivial solutions.