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

This is the authors' abstract. We don't add key points for this paper.

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.