Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with the name of John Tukey, is among the most popular. Tukey's depth has found applications in robust statistics, graph theory, and the study of elections and social choice. We present improved performance guarantees for empirical Tukey's median, a deepest point associated with the given sample, when the data-generating distribution is elliptically symmetric and possibly anisotropic. Some of our results remain valid in the class of affine equivariant estimators. As a corollary of our bounds, we show that the diameter of the set of all empirical Tukey's medians scales like $o(n^{-1/2})$ where $n$ is the sample size.

翻译：是否存在一种自然的方式对高维（维度大于一）数据进行排序？基于数据深度概念的方法（通常与John Tukey的名字相关联）是最流行的方案之一。Tukey深度已在稳健统计学、图论以及选举与社会选择研究中得到广泛应用。本文针对经验Tukey中位数（即与给定样本相关联的最深点）提出了改进的性能保证，其中数据生成分布为椭圆对称分布且可能具有各向异性。我们的部分结果在仿射同变估计量类别中仍然有效。作为这些界限的推论，我们证明所有经验Tukey中位数集合的直径缩放阶为$o(n^{-1/2})$，其中$n$为样本量。