Continuity of halfspace depth contours and maximum depth estimators: Diagnostics of depth-related methods

Continuity of procedures based on the halfspace (Tukey) depth (location and regression setting) is investigated in the framework of continuity concepts from set-valued analysis. Investigated procedures are depth contours (upper level sets) and maximum depth estimators. Continuity is studied both as the pointwise continuity of data-analytic functions, and the weak continuity of statistical functionals-the latter having relevance for qualitative robustness. After a real-data example, some general criteria and counterexamples are given, as well as positive results holding for "typical". data. Finally, some consequences for diagnostics and practical use of the depth-based techniques are drawn. (C) 2002 Elsevier Science (USA).