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 that applies to tail-dependent law-invariant convex risk measures on Orlicz spaces. This concept of robustness captures the tradeoff between robustness and sensitivity and can be quantified by an index of qualitative robustness. By means of this index, we can compare various risk measures, such as distortion risk measures, in regard to their degree of robustness. Our analysis also yields results that are of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for ψ-weak convergence.MSC classification: 62G35, 60B10, 60F05, 91B30, 28A33 JEL Classification D81

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a b s t r a c tThe classical functional delta method (FDM) provides a convenient tool for deriving the asymptotic distribution of statistical functionals from the weak convergence of the respective empirical processes. However, for many interesting functionals depending on the tails of the underlying distribution this FDM cannot be applied since the method typically relies on Hadamard differentiability w.r.t. the uniform sup-norm. In this article, we present a version of the FDM which is suitable also for nonuniform sup-norms, with the outcome that the range of application of the FDM enlarges essentially. On one hand, our FDM, which we shall call the modified FDM, works for functionals that are ''differentiable'' in a weaker sense than Hadamard differentiability. On the other hand, it requires weak convergence of the empirical process w.r.t. a nonuniform sup-norm. The latter is not problematic since there exist strong respective results on weighted empirical processes obtained by Shorack and Wellner (1986) [25], Shao and Yu (1996) [23], Wu (2008) [32], and others. We illustrate the gain of the modified FDM by deriving the asymptotic distribution of plug-in estimates of popular risk measures that cannot be treated with the classical FDM.

It is commonly acknowledged that V-functionals with an unbounded kernel are not Hadamard differentiable and that therefore the asymptotic distribution of U-and V-statistics with an unbounded kernel cannot be derived by the Functional Delta Method (FDM). However, in this article we show that V-functionals are quasi-Hadamard differentiable and that therefore a modified version of the FDM (introduced recently in (J. Multivariate Anal. 101 (2010) 2452-2463)) can be applied to this problem. The modified FDM requires weak convergence of a weighted version of the underlying empirical process. The latter is not problematic since there exist several results on weighted empirical processes in the literature; see, for example, (J. 313-333). The modified FDM approach has the advantage that it is very flexible w.r.t. both the underlying data and the estimator of the unknown distribution function. Both will be demonstrated by various examples. In particular, we will show that our FDM approach covers mainly all the results known in literature for the asymptotic distribution of U-and V-statistics based on dependent data -and our assumptions are by tendency even weaker. Moreover, using our FDM approach we extend these results to dependence concepts that are not covered by the existing literature.

Many standard estimators such as several maximum likelihood estimators or the empirical estimator for any law-invariant convex risk measure are not (qualitatively) robust in the classical sense. However, these estimators may nevertheless satisfy a weak [13,14] or a local [22] robustness property on relevant sets of distributions. One aim of our paper is to identify sets of local robustness, and to explain the benefit of the knowledge of such sets. For instance, we will be able to demonstrate that many maximum likelihood estimators are robust on their natural parametric domains. A second aim consists in extending the general theory of robust estimation to our local framework. In particular we provide a corresponding Hampel-type theorem linking local robustness of a plug-in estimator with a certain continuity condition.