发表机构
The Institute for Data Innovation in Science, Seoul National University; Department of Statistics and School of Transdisciplinary Innovations, Seoul National University; Department of Statistics, Seoul National University（首尔大学数据创新科学研究所; 首尔大学统计学院与跨学科创新中心; 首尔大学统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文指出在二次Wasserstein空间中，常用的局部熵条件在收缩球半径趋于零时失效，即使限制在Lipschitz函数类中亦然，故需新的渐近分析方法。
AI 中文摘要
熵条件广泛用于经验过程分析中，以建立M估计量的渐近性质。在响应变量取值于一般度量空间的回归问题中，一个常见的条件是：以目标对象为中心的收缩球的熵积分在球半径趋于零时保持一致有界。我们证明，在紧区间上支撑的单变量概率分布的二次Wasserstein空间中，该条件通常不成立。具体而言，当目标对象是严格递增且绝对连续的分布函数，且其导数有界且远离零和无穷时，相应的熵积分随着球半径趋于零而发散。即使将环境空间限制为满足固定一致双侧Lipschitz界的分布函数类，这种发散仍然存在。这些结果表明，对于二次Wasserstein空间，需要不同的渐近分析。
英文摘要
Entropy conditions are widely used in empirical-process analyses to establish asymptotic properties of M-estimators. In regression problems with responses taking values in a general metric space, a commonly imposed condition requires the entropy integral associated with a shrinking ball centered at the target object to remain uniformly bounded as the ball radius tends to zero. We show that this condition generally fails in the quadratic Wasserstein space of univariate probability distributions supported on a compact interval. Specifically, when the target object is a strictly increasing and absolutely continuous distribution function whose derivative is bounded away from zero and infinity, the corresponding entropy integral diverges as the ball radius tends to zero. The same divergence persists even when the ambient space is restricted to the class of distribution functions satisfying fixed uniform two-sided Lipschitz bounds. These results indicate that different asymptotic analyses are required for the quadratic Wasserstein space.
