M-estimation with e-statistics

Hongjian Wang; Aaditya Ramdas

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arXiv · 2605.30485

M-estimation with e-statistics

Hongjian Wang·Aaditya Ramdas

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Abstract

We present a theory of point estimation with e-statistics (e-values and e-processes) by introducing the "ME-estimator": the parameter that minimizes the corresponding e-statistic, or the evidence against it. Our approach is based on the intuitive idea of e-statistics as a measure of evidence and betting pay-off, and naturally generalizes the classical method of maximum likelihood estimation. First, we establish the consistency as well as the almost sure convergence rate for ME-estimators relating to the high-probability bounds on the size of the confidence set derived from thresholding the e-statistics, an approach that sets ME-estimators apart from traditional M-estimators. Second, we conduct classical M-estimator-style analysis on the consistency and asymptotic normality of ME-estimators in the bounded mean estimation setting, discussing the notion of efficiency (or lack thereof) from various choices of betting strategy. Our work brings e-statistics, a fundamental tool for inference and uncertainty quantification, to the space of estimation.

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Hongjian Wang, Aaditya Ramdas. 2026-05-28. M-estimation with e-statistics. https://arxiv.org/abs/2605.30485

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