Volume 31, Issue 3-4

We characterize when a convex risk measure associated to a law-invariant acceptance set in L ∞ can be extended to L p , 1≤p<∞$1\le p&lt;\infty $, preserving finiteness and continuity . This problem is strongly connected to the statistical robustness of the corresponding risk measures. Special attention is paid to concrete examples including risk measures based on expected utility, max-correlation risk measures, and distortion risk measures.

In this paper, we study an inference problem for the regression coefficients in some multivariate regression models with multiple change-points occurring at unknown times, when the regression coefficients may satisfy some restrictions. The hypothesized restriction is more general than that given in recent literature. Under a weaker assumption than that given in recent literature, we derive the joint asymptotic normality of the restricted and unrestricted estimators. Finally, we construct a test for the hypothesized restriction and derive its asymptotic power.

In our previous work, we have extended the classical notion of increasing convex stochastic dominance relation with respect to a probability to the more general case of a normalized monotone (but not necessarily additive) set function, also called a capacity. In the present paper, we pursue that work by studying the set of monetary risk measures (defined on the space of bounded real-valued measurable functions) satisfying the properties of comonotonic additivity and consistency with respect to the generalized stochastic dominance relation. Under suitable assumptions on the underlying capacity space, we characterize that class of risk measures in terms of Choquet integrals with respect to a distorted capacity whose distortion function is concave. Kusuoka-type characterizations are also established. A generalization to the case of a capacity of the Tail Value at Risk is provided as an example. It is also shown that some well-known results about Choquet integrals with respect to a distorted probability do not necessarily hold true in the more general case of a distorted capacity.

In this paper, we study the change point test for the tail index of scale-shifted processes. To this task, we propose two tests. The first is designed via examining the discrepancy between the two Hill estimators obtained from the observations before and after a preliminary change point estimate. The second is a modified recursive test which uses scale-adjusted observations. Both methods produce a tail index estimator that outperforms the Hill estimator. A simulation study and real data analysis are provided for illustration.

In this paper, we formulate the classical optimal risk allocation problem for convex risk functionals defined on products of real Banach spaces as risk domains. This generality includes in particular the classical case of L p risks but also allows to describe the influence of dependence in the risk allocation problem. We characterize optimal allocations and complete known existence and uniqueness results from the literature. We discuss in detail an application to expected risk functionals. This case can be dealt with by the Banach space approach applied to Orlicz hearts associated to the risk functionals. We give a detailed discussion of the necessary continuity and differentiability properties. Based on ordering results for Orlicz hearts we obtain extensions of the optimal allocation results to different Orlicz hearts as domain of risk functionals and establish a general form of the classical Borch theorem. In some numerical examples, optimal redistributions are determined for the expected risk case and the precision of the numerical calculation is checked.