On an order-based multivariate median

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.