Continuity of halfspace depth contours and maximum depth estimators: Diagnostics of depth-related methods. (English) Zbl 1028.62042
Summary: 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. The 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.
MSC:
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 62-07 | Data analysis (statistics) (MSC2010) |
| 26E25 | Set-valued functions |
| 54C60 | Set-valued maps in general topology |
| 62H99 | Multivariate analysis |
| 62G35 | Nonparametric robustness |
Keywords:
depth; depth contours; maximum depth estimators; continuity; qualitative robustness; stability; set-valued analysisReferences:
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