地球在移动，偏差也在移动：Wasserstein（地球移动者）距离及基于排列的零校准的系统性向上偏差
The Earth Moves, But So Does the Bias: Systematic Upward Bias of the Wasserstein (Earth Mover's) Distance and Permutation-Based Null Calibration
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中文总结 AI 辅助
本文指出Wasserstein（EMD）距离存在系统性向上偏差，提出基于排列的非参数零校准框架，通过蒙特卡洛模拟验证其效用，可稳健处理偏差问题，适用于度量空间上两概率分布的经验比较。
中文摘要 AI 辅助
地球移动者距离（EMD）在政治科学家中越来越受关注，用于评估偏好分布的相似性。然而，经验概率测度的抽样变异会导致有限样本向上偏差，现有研究对此认识不足。该问题在高维或稀疏场景中尤为严重，包括本文作为示例的联合分布。随着政治科学家拓展EMD的应用场景，本文呼吁该学科需稳健处理向上偏差，提醒勿将标准自助法不确定性区间视为经验EMD向上偏差的校正，提出一种基于排列的零校准框架用于更稳健的假设检验。作为非参数方法，它无需研究者作出方向性或分布形状假设，而其他校正向上偏差的估计量需这些严格假设，政治学数据在实践中常无法满足。本文通过四组蒙特卡洛模拟验证该框架的效用，所提方法也可更广泛应用于共同度量空间上两个概率分布的经验比较，只要支撑点间的地面距离具有实质意义。
英文摘要
The Earth Mover's Distance (EMD) is gaining increasing interest among political scientists for assessing similarity in preference distributions. However, there remains a risk of finite-sample upward bias induced by sampling variation in empirical probability measures, which is under-recognized by existing studies. This problem is especially severe in high-dimensional or sparse settings, including conjoint distributions that serve as an illustrative example in this paper. As political scientists are broadening their use cases of EMD, this paper cautions against interpreting standard bootstrap uncertainty bounds as a correction for the upward bias of empirical EMD. It proposes a permutation-based null calibration framework for more robust hypothesis testing. As a non-parametric approach, it frees researchers from making directional or distributional shape assumptions. While alternative estimators require these rigid assumptions to correct for upward bias, political science data often fail to meet them in practice. Through four sets of Monte Carlo simulations, this paper demonstrates the utility of this framework. The proposed approach also applies more generally to empirical comparisons of two probability distributions defined on a common metric space, provided that the ground distance between support points is substantively meaningful.
