Improved E-Value Thresholds with Applications to Multiple and Sequential Testing
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→0, 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.
Related
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BibTeX
@misc{shen2026improved,
title = {{Improved E-Value Thresholds with Applications to Multiple and Sequential Testing}},
author = {Jiayi Shen and Houzhen Liu and Qirun Zeng and Manhin Poon and Weiqi Yang and Qixin Zhang and Weiwei Zhuang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11186},
eprint = {2610.11186},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11186},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Improved E-Value Thresholds with Applications to Multiple and Sequential Testing
AU - Jiayi Shen
AU - Houzhen Liu
AU - Qirun Zeng
AU - Manhin Poon
AU - Weiqi Yang
AU - Qixin Zhang
AU - Weiwei Zhuang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11186
UR - https://arxiv.org/abs/2610.11186
AB - 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→0, 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.
ER -