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...

When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel’s classical notion of qualitative robustness is not suitable for risk measurement, and we propose and analyze a refined notion of robustness...

A simple and commonly used method to approximate the total claim distribution of a (possibly weakly dependent) insurance collective is the normal approximation. In this article, we investigate the error made when the normal approximation is plugged in a fairly general distribution-invariant risk...

This paper tries to develop a neat and comprehensive probability theory for sample spaces where the events are fuzzy subsets of [InlineMediaObject not available: see fulltext.] The investigations are focussed on the discussion how to equip those sample spaces with suitable σ-algebras and...

Artzner et al. [1] initiated a new direction to assess risks of financial positions by an axiomatic approach. It relies fundamentally on the concept of risk measures, which are functionals defined on sets of financial positions and satisfying some basic properties. The convex risk measures are...

Linear regression models with vague concepts extend the classical single equation linear regression models by admitting observations in form of fuzzy subsets instead of real numbers. They have recently been introduced [cf. Krätschmer, Induktive statistik auf basis unscharfer meßkonzepte am...

Let W denote a family of probability distributions with parameter space [Gamma], and WG be a subfamily of W depending on a mapping G:[Theta]--[Gamma]. Extremum estimations of the parameter vector [theta][set membership, variant][Theta] are considered. Some sufficient conditions are presented to...

Linear regression models with vague concepts extend the classical single equation linear regression models by admitting observations in form of fuzzy subsets instead of real numbers. They have lately been introduced (cf. [V. Krätschmer, Induktive Statistik auf Basis unscharfer Meßkonzepte am...