When estimating the risk of a financial position with empirical data or Monte Carlo simulations via a tail-dependent law invariant risk measure such as the Conditional Value-at-Risk (CVaR), it is important to ensure the robustness of the plug-in estimator particularly when the data contain noise. Krätschmer et al. [Comparative and qualitative robustness for law invariant risk measures. Financ. Stoch., 2014, 18, 271–295.] propose a new framework to examine the qualitative robustness of such estimators for the tail-dependent law invariant risk measures on Orlicz spaces, which is a step further from an earlier work by Cont et al. [Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance, 2010, 10, 593–606] for studying the robustness of risk measurement procedures. In this paper, we follow this stream of research to propose a quantitative approach for verifying the statistical robustness of tail-dependent law invariant risk measures. A distinct feature of our approach is that we use the Fortet–Mourier metric to quantify variation of the true underlying probability measure in the analysis of the discrepancy between the law of the plug-in estimator of the risk measure based on the true data and the one based on perturbed data. This approach enables us to derive an explicit error bound for the discrepancy when the risk functional is Lipschitz continuous over a class of admissible sets. Moreover, the newly introduced notion of Lipschitz continuity allows us to examine the degree of robustness for tail-dependent risk measures. Finally, we apply our quantitative approach to some well-known risk measures to illustrate our results and give an example of the tightness of the proposed error bound.

在使用经验数据或蒙特卡洛模拟通过诸如条件风险价值 (CVaR) 等尾部相关的法律不变风险度量来估计财务状况的风险时，确保插件估计器的稳健性很重要特别是当数据包含噪声时。Krätschmer等人。[法律不变风险措施的比较和定性稳健性。财务。斯托克。, 2014, 18 , 271–295.] 提出了一个新框架来检验这种估计量在 Orlicz 空间上的尾依赖律不变风险度量的定性稳健性，这比 Cont等人的早期工作更进了一步。[风险测量程序的稳健性和敏感性分析。数量。金融, 2010, 10, 593–606] 用于研究风险测量程序的稳健性。在本文中，我们遵循这一研究流，提出了一种量化方法，用于验证尾依赖律不变风险度量的统计稳健性。我们的方法的一个显着特点是我们使用 Fortet-Mourier 度量来量化真实潜在概率度量的变化，以分析基于真实数据的风险度量的插件估计器定律与一种基于扰动数据。当风险函数在一类可容许集合上是 Lipschitz 连续的时，这种方法使我们能够为差异推导出一个明确的错误界限。此外，新引入的 Lipschitz 连续性概念使我们能够检查尾部依赖风险度量的稳健程度。最后，