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

We derive explicitly the e-values with optimal (relative) growth rate in the worst case for testing the mean of a bounded random variable, thereby providing the first application of the (RE)GROW quality criteria beyond the assumption of mutually absolutely continuous hypotheses for e-values originally proposed by Grünwald et al. (2024). For both criteria, we explicitly characterise the alternatives that are most difficult to test against and show that they admit a meaningful interpretation. We give two important examples in which REGROW provides a powerful quality criterion to choose optimal e-variables whereas GROW leads to trivial solutions.