Quasi-Hadamard differentiability of general risk functionals and its application

Krätschmer Volker (), Schied Alexander () and Zähle Henryk ()
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Krätschmer Volker: Faculty of Mathematics, University of Duisburg–Essen, Germany
Schied Alexander: Department of Mathematics, University of Mannheim, Germany
Zähle Henryk: Department of Mathematics, Saarland University, Germany

Statistics & Risk Modeling, 2015, vol. 32, issue 1, 25-47

Abstract: We apply a suitable modification of the functional delta method to statistical functionals that arise from law-invariant coherent risk measures. To this end we establish differentiability of the statistical functional in a relaxed Hadamard sense, namely with respect to a suitably chosen norm and in the directions of a specifically chosen “tangent space”. We show that this notion of quasi-Hadamard differentiability yields both strong laws and limit theorems for the asymptotic distribution of the plug-in estimators. Our results can be regarded as a contribution to the statistics and numerics of risk measurement and as a case study for possible refinements of the functional delta method through fine-tuning the underlying notion of differentiability.

Keywords: Functional delta method; quasi-Hadamard derivative; law-invariant coherent risk measure; Kusuoka representation; weak limit theorem; strong limit theorem (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1515/strm-2014-1174

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