Tail dependence is a probability-based concept meant to address the challenge of detecting and modeling the extreme comovements that can be observed in many real-life situations. Huge financial losses for a bank, floods and epidemics are obvious instances of such extreme comovements. Like extreme value theory in the univariate case, tail dependence depends on asymptotic theory. Therefore, the statistical assessment of tail dependence faces exactly the same problem as extreme value theory: a scarcity of extreme event observations. In the field of dependence modeling, copulas have stood out as a tool of singular importance. They are widely used to account for the various dependence structures that can be encountered in real life. In 2009, Genest et al provided a series of tests to achieve copula selection but showed that these tests were not greatly powerful. This is all the more true when it comes to selecting a copula where tail dependence is crucial. In this paper, we suggest the use of tail indexes in order to detect the presence of tail dependence in a given data set and thus improve the process of selecting a copula. Because tail dependence often goes with data scarcity, we focus on this specific issue through an application to operational losses in the banking industry and propose a way to apply the benefits from theory in practice, while being conscious of the boundaries of such a notion.

中文翻译：

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小样本中的尾依赖：从理论到实践

尾部依赖是一个基于概率的概念，旨在解决检测和建模可以在许多现实生活中观察到的极端联动的挑战。银行的巨额财务损失、洪水和流行病就是这种极端联动的明显例子。与单变量情况下的极值理论一样，尾部依赖依赖于渐近理论。因此，尾部依赖的统计评估面临着与极值理论完全相同的问题：极端事件观测的稀缺性。在依赖建模领域，copula 已经成为一种非常重要的工具。它们被广泛用于解释现实生活中可能遇到的各种依赖结构。在2009年，Genest 等人提供了一系列测试来实现 copula 选择，但表明这些测试并不是很强大。在选择尾依赖至关重要的 copula 时，这一点更加真实。在本文中，我们建议使用尾部索引来检测给定数据集中尾部依赖性的存在，从而改进选择 copula 的过程。由于尾部依赖通常伴随着数据稀缺性，我们通过对银行业运营损失的应用来关注这个特定问题，并提出一种将理论收益应用于实践的方法，同时意识到这种概念的边界。我们建议使用尾部索引来检测给定数据集中尾部依赖性的存在，从而改进选择 copula 的过程。由于尾部依赖通常伴随着数据稀缺性，我们通过对银行业运营损失的应用来关注这个特定问题，并提出一种将理论收益应用于实践的方法，同时意识到这种概念的边界。我们建议使用尾部索引来检测给定数据集中尾部依赖性的存在，从而改进选择 copula 的过程。由于尾部依赖通常伴随着数据稀缺性，我们通过对银行业运营损失的应用来关注这个特定问题，并提出一种将理论收益应用于实践的方法，同时意识到这种概念的边界。