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Improved thresholds for e-values

Christopher Blier-Wong, Ruodu Wang

TL;DR

This work analyzes how to improve rejection thresholds for e-values beyond the default $1/\alpha$ by leveraging distributional information about the e-values. It develops sharp bounds under shape constraints (decreasing, unimodal, log-transformed, and log-concave) and introduces the supremum of comonotonic e-values as a method to preserve type-I control while boosting power. Additional contributions include perturbation bounds for robustness, threshold adjustments for stopped e-processes, and preliminary boosting methods for the e-BH procedure under distributional assumptions. Simulation studies demonstrate substantial power gains across tests within Gaussian families, universal inference scenarios, and multiple testing with boosted e-values. The results offer a principled framework to tailor thresholds to observed distributional features, yielding more efficient and powerful hypothesis testing in practice.

Abstract

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.

Improved thresholds for e-values

TL;DR

This work analyzes how to improve rejection thresholds for e-values beyond the default by leveraging distributional information about the e-values. It develops sharp bounds under shape constraints (decreasing, unimodal, log-transformed, and log-concave) and introduces the supremum of comonotonic e-values as a method to preserve type-I control while boosting power. Additional contributions include perturbation bounds for robustness, threshold adjustments for stopped e-processes, and preliminary boosting methods for the e-BH procedure under distributional assumptions. Simulation studies demonstrate substantial power gains across tests within Gaussian families, universal inference scenarios, and multiple testing with boosted e-values. The results offer a principled framework to tailor thresholds to observed distributional features, yielding more efficient and powerful hypothesis testing in practice.

Abstract

The rejection threshold used for e-values and e-processes is by default set to for a guaranteed type-I error control at , 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 , the threshold can roughly be improved (divided) by a factor of for decreasing or unimodal densities, and by a factor of 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.
Paper Structure (36 sections, 18 theorems, 78 equations, 14 figures, 7 tables)

This paper contains 36 sections, 18 theorems, 78 equations, 14 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Thresholds for e-values: preliminaries
  3. E-variables and thresholds for e-tests
  4. Basic properties
  5. Improved thresholds under distributional information
  6. Decreasing and unimodal densities
  7. Distributions of log-transformed e-variables
  8. Log-concave density, survival or distribution functions
  9. Summary
  10. Perturbation bounds for improved thresholds
  11. Supremum of comonotonic e-values
  12. Threshold for comonotonic e-values
  13. Implications in the Gaussian setting
  14. Maximum likelihood e-values
  15. Comonotonic adjustment to Wilcoxon's signed-rank e-test
  16. ...and 21 more sections

Key Result

Lemma 1

For $\alpha \in (0,1)$, the quantity $T_{\alpha}(\mathcal{E}) : = \inf\{t\ge 1: R_{1/t}(\mathcal{E}) \le \alpha\}$ satisfies If $\gamma\mapsto R_{\gamma}(\mathcal{E})$ is continuous, then $T_{\alpha}(\mathcal{E})$ is the smallest real number $t\ge 1$ such that $\mathbb{P}(E \ge t) \le \alpha$ for all $E\in \mathcal{E}$.

Figures (14)

  • Figure 1: Histogram of null knockoffs, which are e-values up to a constant. The left is based on knockoffs where all the nulls in the dataset are true; the right is based on knockoffs where 10% of variables are non-null.
  • Figure 2: Histogram of universal inference e-values and log e-values. The red line is at $1$ on the left plot and $0$ on the right plot, and the density decreases after the red lines.
  • Figure 3: Proof sketch in part (i) of Theorem \ref{['th:dec-uni']}. Here, $f$ and $g_0$ have the same area (probability) exceeding $z$, with $g_0$ having a smaller mean; $g_1$ has mean 1 and larger probability of exceeding $z$ than $g_0$.
  • Figure 4: Comparison of worst-case type-I errors and improved thresholds for decreasing and unimodal densities.
  • Figure 5: Comparison of worst-case type-I errors and improved thresholds for log-transformed e-variables. Results for $\mathcal{E}_{\rm LUS}$ are conservative bounds since we only find an upper bound for $R_\gamma(\mathcal{E}_{\rm LUS})$ and $T_\alpha(\mathcal{E}_{\rm LUS})$.
  • ...and 9 more figures

Theorems & Definitions (39)

  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Proposition 1
  • Theorem 2
  • proof
  • Corollary 1
  • Remark 2
  • ...and 29 more