On Asymptotic Efficiency and Robustness of Multivariate Medians by

Univariate median is a well-known location estimator, which is√ n-consistent, asymptotically Gaussian and affine equivariant. It is also a robust estimator of location with the highest asymptotic break-down point (50%). While there are several versions of multivariate median proposed and extensively studied in the literature, many of those statistical properties of univariate median fail to hold for some of those multivariate medians. In this article, we make a comprehen-sive review of the known robustness and other asymptotic properties of different versions of multivariate median as well as some other well known multivariate location estimators with 50 % breakdown points and derive some new results on asymptotic efficiency of those esti-mators. Our investigation shows that the affine equivariant version of spatial median obtained using the transformation-retransformation technique, which has 50 % breakdown point, has optimal asymptotic properties among the popular versions of multivariate median con-sidered in the literature for data following elliptic distributions with exponential and polynomial tails, which include multivariate normal and multivariate Cauchy distributions as special cases. Further, when compared with some other well-known robust estimates of location, the transformation-retransformation spatial median tends to have su-perior asymptotic performance in a number of cases