On the Precise Asymptotics of Universal Inference

Traditional statistical inference techniques, such as likelihood ratio tests, have seen renewed interest in recent years, driven in part by the growing emphasis on methodologies based on e-values and e-processes, rather than conventional p-values. Unlike p-values, e-values possess several properties that make them particularly appealing for modern data science applications. In particular, e-value-based methods have played an instrumental role in advancing multiple and safe testing (Grünwald et al., 2020; Vovk and Wang, 2021; Shafer, 2021; Wang and Ramdas, 2022), anytime-valid inference (Waudby-Smith and Ramdas, 2024), and asymptotic confidence sequences (Waudby-Smith et al., 2024). This list is far from exhaustive, and we refer to Ramdas et al. (2023) for a broader overview of recent developments. This manuscript revisits the work of Wasserman et al. (2020), who introduced universal inference, a general hypothesis testing framework based on split likelihood ratio statistics, which is also an e-value. This framework provides simple procedures for many complex composite testing problems that previously lacked actionable solutions, such as testing logconcavity (Dunn et al., 2024) and causal inference under unknown causal structures (Strieder et al., 2021), among others. Specifically, universal inference combines the classical idea of sample splitting (Cox, 1975) and Markov's inequality to establish finite-sample validity. The procedure follows three steps.