Improved thresholds for e-values

The rejection threshold used for e-values and e-processes is by default set to 1/α for a guaranteed type-I error control at α, based on Markov’s and Ville’s inequalities. This threshold can be wasteful in practical applications. We discuss how this threshold can be improved under additional distributional assumptions on the e-values; some of these assumptions are naturally plausible and empirically observable, without knowing explicitly the form or model of the e-values. For small values of α, the threshold can roughly be improved (divided) by a factor of 2 for decreasing or unimodal densities, and by a factor of e for decreasing or unimodal-symmetric densities of log-transformed e-values. Moreover, we propose to use the supremum of comonotonic e-values, which is shown to preserve the type-I error guarantee. We also propose some preliminary methods to boost e-values in the e-BH procedure under some distributional assumptions while controlling the false discovery rate. Through a series of simulation studies, we demonstrate the effectiveness of our proposed methods in various testing scenarios, showing enhanced power.