On the statistical efficiency of robust estimators of multivariate location

Abstract

The 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 breakdown point (i.e., 50%). While there are several versions of the multivariate median proposed and extensively studied in the literature, many of the aforesaid statistical properties of the univariate median fail to hold for some of those multivariate medians. Among the multivariate medians, the affine equivariant versions of spatial and coordinatewise medians have 50% asymptotic breakdown point, and they have asymptotically Gaussian distributions. The minimum covariance determinant (MCD) estimator is another widely used robust estimator of multivariate location, which is also affine equivariant, with 50% asymptotic breakdown point, and its asymptotic distribution is Gaussian. In this article, we make a comparative study of the efficiencies of affine equivariant versions of spatial and coordinatewise medians and the efficiencies of the MCD and related estimators considered in the literature.