Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness

We describe multivariate generalizations of the median, trimmed mean and W estimates. The estimates are based on a geometric construction related to "projection pursuit. " They are both affine equivariant (coordinate-free) and have high breakdown point. The generalization of the median has a breakdown point of at least 1/ (d + 1) in dimension d and the breakdown point can be as high as 1/3 under symmetry. In contrast, various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than 1/ (d + 1) in dimension d.