A note on the robustness of multivariate medians

Abstract

In this note we investigate the extent to which some of the fundamental properties of univariate median are retained by different multivariate versions of median with special emphasis on robustness and breakdown properties. We show that transformation retransformation medians, which are affine equivariant, n^½-consistent and asymptotically normally distributed under standard regularity conditions, can also be very robust with high breakdown points. We prove that with some appropriate adaptive choice of the transformation matrix based on a high breakdown estimate of the multivariate scatter matrix (e.g. S-estimate or minimum covariance determinant estimate), the finite sample breakdown point of a transformation retransformation median will be as high as n-1[(n-d+1)/2], where n=the sample size, d=the dimension of the data, and [x] denotes the largest integer smaller than or equal to x. This implies that as n-->[infinity], the asymptotic breakdown point of a transformation retransformation median can be made equal to 50% in any dimension just like the univariate median. We present a brief comparative study of the robustness properties of different affine equivariant multivariate medians using an illustrative example.