Abstract Although the order-based definition of the univariate median is ubiquitous in statistics, the same order-based definition is typically abandoned when extending the univariate median to higher dimensions. In this paper, an example of order-based multivariate median based on the use of a linear extension of the product order is brought to the attention. Symmetry and internality properties are fulfilled by such order-based multivariate median, however, it turns out that this function fails to fulfil in general other appealing properties such as equivariance to different geometrical transformations, monotonicity properties and continuity. Interestingly, translation and scale equivariance and some monotonicity properties can be guaranteed if the linear extension of the product order is carefully chosen. Finally, it is proved that, unlike in the univariate setting, the finite-sample breakdown point of the order-based multivariate median is 1/n, thus implying that it is not robust in the presence of outliers.

中文翻译：

---

基于顺序的多元中位数

摘要 尽管单变量中位数的基于顺序的定义在统计学中无处不在，但在将单变量中位数扩展到更高维度时，通常会放弃相同的基于顺序的定义。在本文中，基于使用乘积阶的线性扩展的基于阶的多元中值的示例引起了注意。这种基于顺序的多元中位数满足了对称性和内部性属性，然而，事实证明，该函数通常无法满足其他吸引人的属性，例如对不同几何变换的等变、单调性属性和连续性。有趣的是，如果仔细选择乘积阶的线性扩展，则可以保证平移和尺度等变以及一些单调性。最后证明，