Qualitative and infinitesimal robustness of tail-dependent statistical functionals |
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Authors: | Volker Krätschmer |
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Institution: | a University of Duisburg-Essen, Faculty of Mathematics, Forsthausweg 2, D-47057 Duisburg, Germanyb University of Mannheim, Department of Mathematics, A5, 6, D-68131 Mannheim, Germanyc Saarland University, Department of Mathematics, Postfach 151150, D-66041 Saarbrücken, Germany |
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Abstract: | The main goal of this article is to introduce a new notion of qualitative robustness that applies also to tail-dependent statistical functionals and that allows us to compare statistical functionals in regards to their degree of robustness. By means of new versions of the celebrated Hampel theorem, we show that this degree of robustness can be characterized in terms of certain continuity properties of the statistical functional. The proofs of these results rely on strong uniform Glivenko-Cantelli theorems in fine topologies, which are of independent interest. We also investigate the sensitivity of tail-dependent statistical functionals w.r.t. infinitesimal contaminations, and we introduce a new notion of infinitesimal robustness. The theoretical results are illustrated by means of several examples including general L- and V-functionals. |
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Keywords: | 62G05 62G35 |
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