Reference-Null Calibrated Thresholds for E-Processes with Applications to Conformal Martingales
Abstract
E-processes provide a flexible framework for anytime-valid inference, with the conventional rejection boundary 1/α typically justified by Ville's inequality. Such a boundary is universal but can be conservative, as it does not exploit additional information about the null distribution. We propose a reference-null calibration framework that uses independent null samples to construct sharper rejection thresholds while preserving type-I error control. To study the statistical gain from sharper thresholds, we specialize the framework to conformal martingales. Building on histogram-based betting, we incorporate Krichevsky--Trofimov smoothing and develop a restart-mixture construction for distribution shift detection. We establish quantitative results for detection power and detection delay and characterize how the reduction in the rejection boundary translates into power and delay gains relative to the conventional Ville boundary. Numerical experiments corroborate the theoretical findings across a range of distributional changes. Finally, we apply the proposed calibration strategy to an existing e-process for online LLM watermark detection, showing that the method can improve sequential detection performance without modifying the underlying e-process. These results demonstrate that reference-null calibration provides a general and modular way to enhance the efficiency of e-process-based sequential inference.
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BibTeX
@misc{ding2026referencenull,
title = {{Reference-Null Calibrated Thresholds for E-Processes with Applications to Conformal Martingales}},
author = {Yi Ding and Lan Wei and Xuehu Zhu and Wenlin Dai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32678},
eprint = {2609.32678},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32678},
}
RIS
TY - UNPB
TI - Reference-Null Calibrated Thresholds for E-Processes with Applications to Conformal Martingales
AU - Yi Ding
AU - Lan Wei
AU - Xuehu Zhu
AU - Wenlin Dai
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32678
UR - https://arxiv.org/abs/2609.32678
AB - E-processes provide a flexible framework for anytime-valid inference, with the conventional rejection boundary 1/α typically justified by Ville's inequality. Such a boundary is universal but can be conservative, as it does not exploit additional information about the null distribution. We propose a reference-null calibration framework that uses independent null samples to construct sharper rejection thresholds while preserving type-I error control. To study the statistical gain from sharper thresholds, we specialize the framework to conformal martingales. Building on histogram-based betting, we incorporate Krichevsky--Trofimov smoothing and develop a restart-mixture construction for distribution shift detection. We establish quantitative results for detection power and detection delay and characterize how the reduction in the rejection boundary translates into power and delay gains relative to the conventional Ville boundary. Numerical experiments corroborate the theoretical findings across a range of distributional changes. Finally, we apply the proposed calibration strategy to an existing e-process for online LLM watermark detection, showing that the method can improve sequential detection performance without modifying the underlying e-process. These results demonstrate that reference-null calibration provides a general and modular way to enhance the efficiency of e-process-based sequential inference.
ER -