The asymptotic efficiency of the spatial median for elliptically symmetric distributions

A detailed study of the asymptotic efficiency of the spatial median under elliptically symmetric distributions is presented. In particular, it is shown that its asymptotic relative efficiency with respect to an affinely equivariant version of the spatial median does not exceed one, and consequently the spatial median is asymptotically inadmissible at elliptical distributions. The degree of the inefficiency of the spatial median depends on the true underlying scatter matrix parameter, with its inefficiency being quite severe, but not always severe, when the scatter matrix is far from being proportional to the identity. Finally, a simulation study is presented for the finite sample relative efficiency of the spatial median. This simulation study reveals some curious oscillating patterns in the efficiencies as the sample size increases.