Multivariate median

Abstract

Multivariate medians are robust competitors of the mean vector in estimating the symmetry center of a multivariate distribution. Various definitions of the multivariate medians have been proposed in the literature, and their properties (efficiency, equivariance, robustness, computational convenience, estimation of their accuracy, etc.) have been extensively investigated. The univariate median as well as the univariate concepts of sign and rank are based on the ordering of the univariate observations. Unfortunately, there is no natural ordering of multivariate data points. An approach utilizing L1objective functions is therefore often used to extend these concepts to the multivariate case. In this contribution we consider three multivariate extensions of the median, the vector of marginal medians, the spatial median, and the Oja median, based on three different multivariate L1objective functions, and review their statistical properties as found in the literature.