标题： 分位数断层扫描：在多变量数据中使用分位数 显示英文标题

标题： Quantile tomography: using quantiles with multivariate data

摘要： 利用分位数来获取关于多变量数据的见解被进行了探讨。 认为通过考虑方向分位数，即投影的分位数，可以获得深刻的见解。 提出了方向分位数包络线作为一种浓缩此类信息的方法；证明它们本质上是半空间（Tukey）深度水平集，在椭圆分布（特别是多变量正态分布）中与密度轮廓重合。 研究了有关它们的索引、反向检索方向分位数信息的可能性、仿射变换下的不变性以及近似/渐近性质的相关问题。 认为基于方向分位数及其包络线的分析提供了直接的概率解释，因此传达了具体的定量意义；方向定义可以适应更复杂的框架，如极端分位数的估计和方向分位数回归，以及深度轮廓对协变量的回归。 后者有助于构建多变量生长图——这一问题促使了所有的发展。 显示英文摘要

摘要： The use of quantiles to obtain insights about multivariate data is addressed. It is argued that incisive insights can be obtained by considering directional quantiles, the quantiles of projections. Directional quantile envelopes are proposed as a way to condense this kind of information; it is demonstrated that they are essentially halfspace (Tukey) depth levels sets, coinciding for elliptic distributions (in particular multivariate normal) with density contours. Relevant questions concerning their indexing, the possibility of the reverse retrieval of directional quantile information, invariance with respect to affine transformations, and approximation/asymptotic properties are studied. It is argued that the analysis in terms of directional quantiles and their envelopes offers a straightforward probabilistic interpretation and thus conveys a concrete quantitative meaning; the directional definition can be adapted to elaborate frameworks, like estimation of extreme quantiles and directional quantile regression, the regression of depth contours on covariates. The latter facilitates the construction of multivariate growth charts---the question that motivated all the development.