The Influence Function of Scatter Halfspace Depth

In this work, we investigate robustness properties of scatter halfspace depth. Its influence function (IF) and sensitivity curve (SC) are studied in the known and unknown location case. In the former, both functions have explicit expressions and are showed to be bounded and discontinuous. In the latter, we provide bounds on the IF that depend explicitly on the location functional used. Similar derivations are provided for the symmetrized scatter halfspace depth based on differences. Likewise, the influence function of the associated scatter halfspace depth median is investigated, with explicit expression in the elliptical setting.