论文标题
尾部依赖性，超级集和度量嵌入
Tail-dependence, exceedance sets, and metric embeddings
论文作者
论文摘要
在随机矢量中测量和建模的尾巴依赖性有很多方法：从多变量规则变化的一般框架以及最大稳定向量的灵活类别到简单明了的简洁摘要措施，例如双变量尾依赖性系数的矩阵。本文首先从统一的角度提供对现有结果的回顾，该角度突出了极值理论与削减和指标理论之间的联系。我们的方法导致在这两个领域中有一些新发现，并在风险管理中进行了一些应用。 我们首先使用多元规则变化的框架，以表明极端系数或同等地，可以简单地理解随机矢量的高阶尾部依赖性系数，以随机超出集合而理解，这使我们能够扩展Bernoulli兼容性的概念。在双变量尾部依赖性的特殊但重要的情况下，我们通过频谱距离建立了尾巴依赖性矩阵与$ l^1 $ - 和$ \ ell_1 $ - ELL_1 $ - ELL_1 $ - ELL_1 $ - ELL_1 $ - ELL_1 $ - ELL_1 $ - ELL_1 $ - ELL_1 $ - ELL_1 $ - ELL_1 $ - EMBEDDABLE TORITE METRITRICT距离。也就是说，光谱距离的切割的系数和Tawn-Molchanov Max-stable模型实现了相应的双变量极端依赖性。我们表明，线指标是刚性的，如果光谱距离对应于线指标，则高阶尾部依赖性由双变量的尾依赖性矩阵确定。 最后，$ \ ell_1 $ embeddable度量空间和尾部依赖性矩阵之间的对应关系使我们能够重新审视可靠性问题，即检查给定矩阵是否是有效的尾巴依赖性矩阵。我们证实了Shyamalkumar＆Tao（2020）的猜想，即此问题是NP完成的。
There are many ways of measuring and modeling tail-dependence in random vectors: from the general framework of multivariate regular variation and the flexible class of max-stable vectors down to simple and concise summary measures like the matrix of bivariate tail-dependence coefficients. This paper starts by providing a review of existing results from a unifying perspective, which highlights connections between extreme value theory and the theory of cuts and metrics. Our approach leads to some new findings in both areas with some applications to current topics in risk management. We begin by using the framework of multivariate regular variation to show that extremal coefficients, or equivalently, the higher-order tail-dependence coefficients of a random vector can simply be understood in terms of random exceedance sets, which allows us to extend the notion of Bernoulli compatibility. In the special but important case of bi-variate tail-dependence, we establish a correspondence between tail-dependence matrices and $L^1$- and $\ell_1$-embeddable finite metric spaces via the spectral distance, which is a metric on the space of jointly $1$-Fréchet random variables. Namely, the coefficients of the cut-decomposition of the spectral distance and of the Tawn-Molchanov max-stable model realizing the corresponding bi-variate extremal dependence coincide. We show that line metrics are rigid and if the spectral distance corresponds to a line metric, the higher order tail-dependence is determined by the bi-variate tail-dependence matrix. Finally, the correspondence between $\ell_1$-embeddable metric spaces and tail-dependence matrices allows us to revisit the realizability problem, i.e. checking whether a given matrix is a valid tail-dependence matrix. We confirm a conjecture of Shyamalkumar & Tao (2020) that this problem is NP-complete.
