Bentkus-type asymptotic e-values

E-values have recently emerged as a versatile alternative to p-values for statistical inference (Ramdas and Wang, 2025). They offer several advantages: they remain valid under optional stopping (Grünwald et al., 2024a), combine easily under arbitrary dependence, and exist for irregular problems where no other inferential method is known (Wasserman et al., 2020), among others. Beyond being useful, they have also proven necessary in various problems, such as multiple testing (Wang and Ramdas, 2022; Fischer and Ramdas, 2024; Xu et al., 2025), statistical contract theory (Bates et al., 2022), and post-hoc inference (Grünwald, 2024). Formally, an e-value is a nonnegative test statistic whose expected value is at most one under the null hypothesis. Ideally, analysts want e-values that are large under the alternative--that is, e-values with high power.