多元分位数：基于几何与测度传输的等高线
Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours
AI总结：
本文比较了多元分位数扩展中的两种主要概念——geometric quantiles与基于measure transportation的center-outward quantiles，分析了它们的主要理论性质与数值差异。
摘要：
分位数是概率论与理论统计学中的基本概念，也是日常应用中的常用工具。虽然一元分位数的概念相当清晰且已被充分理解，但其多元扩展则更为困难。经过半个世纪的持续努力与众多提案，实际上正涌现出两种概念：所谓的（重标记）geometric quantiles，将一元分位数作为涉及check functions的L1损失函数的最小化者的特征进行扩展；以及基于measure transportation思想的较新的center-outward quantiles。这两个概念产生了不同的分位数区域族和分位数等高线族。我们的目标是呈现它们主要理论性质的比较，并对其差异进行数值研究。
英文摘要：
Quantiles are a fundamental concept in probability and theoretical statistics and a daily tool in their applications. While the univariate concept of quantiles is quite clear and well understood, its multivariate extension is more problematic. After half a century of continued efforts and many proposals, two concepts, essentially, are emerging: the so-called (relabeled) geometric quantiles, extending the characterization of univariate quantiles as minimizers of an L1 loss function involving the check functions, and the more recent center-outward quantiles based on measure transportation ideas. These two concepts yield distinct families of quantile regions and quantile contours. Our objective here is to present a comparison of their main theoretical properties and a numerical investigation of their differences.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
