M-estimation with e-statistics

2026-05-28

AI Summary

This work introduces e-statistics into parametric point estimation and proposes the minimum e-statistic (ME) estimator, which estimates parameters by minimizing the e-statistic—interpreted as evidence or betting payoff—against them, thereby offering a natural generalization of maximum likelihood estimation. The approach overcomes limitations of classical M-estimation regarding evidence quantification and confidence set construction. We establish, for the first time, theoretical guarantees for the ME estimator, including consistency, almost sure convergence rates, and asymptotic normality, and construct high-probability confidence sets via thresholded e-statistics. Furthermore, we systematically analyze how different betting strategies influence estimation efficiency, successfully extending the applicability of e-statistics from hypothesis testing to point estimation.

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.

Problem

• e-statistics
• point estimation
• M-estimation
• uncertainty quantification
• inference

Innovation

• e-statistics
• ME-estimator
• M-estimation
• asymptotic normality
• confidence sets