Wenkai Xu’s contribution to the Discussion of ‘Safe Testing’ by Peter Grünwald, Rianne de Heide, and Wouter M. Koolen

The authors propose the framework of e-value testing by investigating the grow-rate-optimal (GRO) constructions of e-variables, that incorporate a comprehensive range of testing scenarios for simple alternative, as well as their composite counterparts.The authors illustrate the versatility of e-value tests via three interpretations, drawing connections to 1) game-theoretical viewpoint, 2) conservative pvalues, and 3) Bayes Factors; and devise the determination and connections for optional continuation (OC) and optional stopping (OS) with e-variables, going beyond the capacity of p-value based testing.The proposed GRO-based tests (and the corresponding variants) also embed intriguing connections with information processing, that strengthen the proposal on testing with streaming.