Improved E-Value Thresholds with Applications to Multiple and Sequential Testing

2026-10-07

AI Summary

This study addresses the excessive conservativeness of universal e-value thresholds under structured null distributions by proposing a calibration method based on non-decreasing and Lipschitz density models. Theoretically, we derive asymptotically optimal Lipschitz-calibrated thresholds by leveraging Markov inequality constraints, minimax theory, and conditional e-value construction techniques. Practically, this approach is extended to false discovery rate (FDR) control and sequential analysis settings. The proposed method substantially increases the number of discoveries in multiple testing scenarios. Furthermore, it demonstrates a favorable trade-off between statistical power and threshold sharpness within sequential procedures. Overall, this work provides a more efficient solution for statistical inference under structured null hypotheses.

Abstract

E-values provide a flexible framework for statistical inference, but the universal threshold $1/α$ can be conservative when the null distribution has additional structure. We study how structural restrictions limit the worst-case concentration underlying Markov's inequality, using a nondecreasing density model to constrain tail allocation and an $L$-Lipschitz density model to control local concentration. The nondecreasing model gives the minimax rejection threshold, and the Lipschitz condition yields a sharper closed-form threshold with an $O(L^{-1/2})$ relative improvement over $1/α$. We further show that this threshold is asymptotically sharp as $α\to0$, in the sense that no uniformly valid threshold can improve on it by a fixed positive $L$-dependent amount. We then incorporate the Lipschitz calibration into multiple testing procedures with FDR control and use the same structural information to construct calibrated conditional e-values for anytime-valid sequential inference. Numerical experiments illustrate the resulting gains in discoveries and the trade-offs of the sequential procedures.

Problem

• E-values
• threshold calibration
• multiple testing
• sequential inference
• Markov's inequality

Innovation

• E-values
• Lipschitz density model
• multiple testing
• sequential inference
• minimax threshold