Robustness of weighted Lp–depth and Lp–median

Lp–norm weighted depth functions are introduced and the local and global robustness of these weighted Lp–depth functions and their induced multivariate medians are investigated via influence function and finite sample breakdown point. To study the global robustness of depth functions, a notion of finite sample breakdown point is introduced. The weighted Lp–depth functions turn out to have the same low breakdown point as some other popular depth functions. Their influence functions are also unbounded. On the other hand, the weighted Lp–depth induced medians are globally robust with the highest possible breakdown point for any reasonable estimator. The weighted Lp–medians are also locally robust with bounded influence functions for suitable weight functions. Unlike other existing depth functions and multivariate medians, the weighted Lp depth and medians are easy to calculate in high dimensions. The price for this advantage is the lack of affine invariance and equivariance of the weighted Lp depth and medians, respectively.