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.

翻译：分位数是概率论与理论统计学中的基本概念，也是其应用中的日常工具。虽然单变量分位数的概念清晰且易于理解，但其多元扩展却更具挑战性。经过半个世纪的持续努力与众多提议，目前主要涌现出两个概念：一是（重新命名的）几何分位数，它扩展了单变量分位数作为涉及检查函数的L1损失函数极小值点的特征；二是基于测度传输思想的最新中心向外分位数。这两个概念产生了截然不同的分位数区域与分位数轮廓族。本文旨在比较它们的主要理论性质，并通过数值研究探讨其差异。