In this paper we investigate the robustness properties of the deepest regression, a method for linear regression introduced by Rousseeuw and Hubert [6]. We show that the deepest regression functional is Fisher-consistent for the conditional median, and has a breakdown value of in all dimensions....

For multivariate data, the halfspace depth function can be seen as a natural and affine equivariant generalization of the univariate empirical cdf. For any multivariate data set, we show that the resulting halfspace depth function completely determines the empirical distribution. We do this by...