The definition of multivariate quantiles has gained considerable attention in previous years as a tool for understanding the structure of a multivariate data cloud. Due to the lack of a natural ordering for multivariate data, many approaches have either considered geometric generalisations of univariate quantiles or data depths that measure centrality of data points. Both approaches provide a centre-outward ordering of data points but do no longer possess a relation to the cumulative distribution function of the data generating process and corresponding tail probabilities. We propose a new notion of bivariate quantiles that is based on inverting the bivariate cumulative distribution function and therefore provides a directional measure of extremeness as defined by the contour lines of the cumulative distribution function which define the quantile curves of interest. To determine unique solutions, we transform the bivariate data to the unit square. This allows us to introduce directions along which the quantiles are unique. Choosing a suitable transformation also ensures that the resulting quantiles are equivariant under monotonically increasing transformations. We study the resulting notion of bivariate quantiles in detail, with respect to computation based on linear programming and theoretical properties including asymptotic behaviour and robustness. It turns out that our approach is especially useful for data situations that deviate from the elliptical shape typical for ‘normal-like’ bivariate distributions. Moreover, the bivariate quantiles inherit the robustness of univariate quantiles even in case of extreme outliers.

中文翻译：

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定向双变量分位数：基于累积分布函数的稳健方法

近年来，作为理解多元数据云结构的工具，多元分位数的定义引起了广泛的关注。由于缺乏对多元数据的自然排序，因此许多方法要么考虑了单变量分位数的几何概括，要么考虑了测量数据点中心性的数据深度。两种方法都提供了数据点的中心向外排序，但不再与数据生成过程的累积分布函数和相应的尾部概率有关。我们提出了一种新的双变量分位数概念，该概念基于对双变量累积分布函数的求逆，因此提供了一种由累积分布函数轮廓线定义的极限方向性度量，该轮廓线定义了所关注的分位数曲线。为了确定唯一的解决方案，我们将双变量数据转换为单位平方。这使我们能够介绍分位数唯一的方向。选择合适的变换还可以确保在单调递增的变换下，得到的分位数相等。关于基于线性规划和理论特性（包括渐近行为和鲁棒性）的计算，我们详细研究了二元分位数的结果概念。事实证明，对于偏离“正态”双变量分布典型的椭圆形状的数据情况，我们的方法特别有用。而且，即使在极端离群值的情况下，双变量分位数也继承了单变量分位数的鲁棒性。