Improved thresholds for e-values

TL;DR

This paper proposes improved thresholds for e-values in hypothesis testing, leveraging distributional assumptions to enhance power while maintaining error control, and introduces methods to boost e-values in multiple testing procedures.

Contribution

It introduces new thresholding techniques for e-values based on distributional assumptions and proposes methods to improve e-value boosting in multiple testing.

Findings

Improved thresholds can roughly double or e-fold reduce the standard threshold.

Supremum of comonotonic e-values preserves type-I error control.

Simulation studies show enhanced power in various testing scenarios.

Abstract

The rejection threshold used for e-values and e-processes is by default set to $1/\alpha$ for a guaranteed type-I error control at $\alpha$, 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 $\alpha$, 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.