多元分布的quantiles：总体概念的短综述
Quantiles for multivariate distribution: a short survey of population concepts
中文总结 AI 辅助
本文综述了多元分布中quantiles的总体概念，指出一维定义依赖实数序而高维缺乏典范序，总结了多种扩展定义并更新了Serfling的经典综述。
中文摘要 AI 辅助
Quantiles是概率论与统计学中最基本的概念之一，从描述性统计到推断统计均如此。然而，quantile函数在一维概率分布的背景下定义明确且被充分理解，其定义是分布函数的逆——这一定义与实数轴上的典范序密切相关。从维度$d=2$开始，在${\mathbb R}^d$中不再存在这样的典范序；在诸如超球面、环面或多环面（方向性变量的向量）等非线性流形中也不存在。这导致缺乏一个明显且被广泛接受的quantiles定义。尽管如此，将quantile概念扩展到经典单变量背景之外的需求催生了大量文献，产生了多种或多或少令人满意的quantile概念；近年来在这一方面尤为活跃。本短综述旨在补充和更新Serfling 25年前的综述（Serfling 2008），通过总结并统一其中一些概念，并结合近期贡献。
英文摘要
Quantiles are among the most fundamental concepts in Probability and Statistics, from descriptive to inferential. However, quantile functions are well-defined and well-understood in the context of one-dimensional probability distributions, where their definition as the inverse of distribution functions---a definition which is intimately related to the canonical ordering of the real line. Starting with dimension $d=2$, such a canonical ordering is no longer available in~${\mathbb R}^d$; nor is it available in nonlinear manifolds such as hyperspheres, tori, or polyspheres (vectors of directional variables). This results in the absence of an obvious and widely accepted definition of quantiles. The need to extend the concept of quantile beyond the classical univariate context nevertheless has sparked a large body of literature, giving rise to a variety of more or less satisfactory quantile concepts; the recent years have been particularly active in this respect. This short review is an attempt to complement and update Serfling's 25-year-old survey (Serfling 2008) by summarizing and unifying some of these concepts in light of recent contributions.
