标题： 重新思考均方误差：为什么信息是评估估计量的更优标准 显示英文标题

标题： Rethinking Mean Square Error: Why Information is a Superior Assessment of Estimators

摘要： 詹姆斯-斯坦估计量在均方误差（MSE）方面优于最大似然估计，是现代统计学中最受赞誉的结果之一，表明有偏估计量可以系统地优于无偏估计量。 我们认为这一结论源于使用了不恰当的评估标准。 通过简单的模拟，我们证明了虽然詹姆斯-斯坦估计量具有较低的MSE，但在实践中却表现出令人担忧的行为：基于詹姆斯-斯坦的假设检验可能具有低于显著性水平的效力，表现出严重的不对称性，并导致从业者不愿报告的结论。 使用$\Lambda$信息（Vos和Wu，2025），一种衡量估计量区分不同分布效果的标准，我们表明最大似然估计达到了完全效率，而詹姆斯-斯坦估计量在MSE表明其优势的地方表现却很差。 我们的分析揭示了MSE的根本缺陷——以点为基础评估估计量，从而忽略了估计的重要方面——造成了这些悖论。 通过从点估计量扩展到广义估计量（参数空间上的函数），我们获得了一个参数不变的框架，统一了估计和检验。 这些见解表明，统计界应重新考虑的不是最大似然理论，而是我们对MSE用于比较估计量的依赖。 显示英文摘要

摘要： The James-Stein estimator's dominance over maximum likelihood in terms of mean square error (MSE) has been one of the most celebrated results in modern statistics, suggesting that biased estimators can systematically outperform unbiased ones. We argue that this conclusion stems from using an inappropriate assessment criterion. Through simple simulations, we demonstrate that while James-Stein achieves lower MSE, it produces concerning behavior in practice: hypothesis tests based on James-Stein can have power below the significance level, exhibit severe asymmetry, and lead to conclusions that practitioners would hesitate to report. Using $\Lambda$-information (Vos and Wu, 2025), a criterion that measures how effectively estimators distinguish between distributions, we show that maximum likelihood achieves full efficiency while James-Stein performs poorly precisely where MSE suggests superiority. Our analysis reveals that MSE's fundamental flaw--assessing estimators point-wise thereby missing important aspects of estimation--creates these paradoxes. By expanding from point estimators to generalized estimators (functions over the parameter space), we obtain a parameter-invariant framework that unifies estimation and testing. These insights suggest that the statistical community should reconsider not maximum likelihood theory, but rather our reliance on MSE for comparing estimators.