While the estimation of risk is an important question in the daily business of banks and insurances, many existing plug-in estimation procedures suffer from an unnecessary bias. This often leads to the underestimation of risk and negatively impacts backtesting results, especially in small sample cases. In this article we show that the link between estimation bias and backtesting can be traced back to the dual relationship between risk measures and the corresponding performance measures, and discuss this in reference to value-at-risk and expected shortfall frameworks. Motivated by this finding, we propose a new algorithm for bias correction and show how to apply it for generalized Pareto distributions. In particular, we consider value-at-risk and expected shortfall plug-in estimators, and show that the application of our algorithm leads to gain in efficiency when heavy tails exist in the data.

翻译：虽然风险估算是银行和保险业日常业务中的一个重要问题,但许多现有的插头估算程序都存在不必要的偏差,这往往导致低估风险和消极影响,特别是小型抽样案例的反向测试结果;在本篇文章中,我们表明,估计偏差和反向测试之间的联系可以追溯到风险措施与相应的绩效措施之间的双重关系,并讨论这一点,并参照风险价值和预期的短缺框架。受这一发现的影响,我们提出了新的偏差纠正算法,并表明如何将其应用于泛泛Pareto的分布。特别是,我们考虑了风险价值和预期缺额估测值,并表明当数据中存在重尾巴时,应用我们的算法会提高效率。