On the Effect of Inliers on the Spatial Median

We point out that inliers adversely affect performance of the spatial median and its generalization due to Gentleman. They are most deleterious in the case of the median itself, and in the important setting of two dimensions. There, the second term in a stochastic expansion of the median has a component with a Cauchy limiting distribution, and does not have any finite moments. This term is substantially determined by a small number of extreme, inlying data values. The implications for bootstrap methods are significant, since the bootstrap is notoriously poor in capturing properties of extremes. Indeed, the bootstrap does not accurately approximate second-order features of the distribution of the two-dimensional spatial median. We suggest a Winsorizing device for alleviating the effects of inliers. The issue of outliers is also discussed.