Among bivariate tail dependence measures, the tail dependence coefficient has emerged as the popular choice. Akin to the correlation matrix, a multivariate dependence measure is constructed using these bivariate measures, and this is referred to in the literature as the tail dependence matrix (TDM). While the problem of determining whether a given d × d matrix is a correlation matrix is of the order O(d³) in complexity, determining if a matrix is a TDM (the realization problem) is believed to be non-polynomial in complexity. Using a linear programming (LP) formulation, we show that the combinatorial structure of the constraints is related to the intractable max-cut problem in a weighted graph. This connection provides an avenue for constructing parametric classes admitting a polynomial in d algorithm for determining membership in its constraint polytope. The complexity of the general realization problem is justifiably of much theoretical interest. Since in practice one resorts to lower dimensional parametrization of the TDMs, we posit that it is rather the complexity of the realization problem restricted to parametric classes of TDMs, and algorithms for it, that are more practically relevant. In this paper, we show how the inherent symmetry and sparsity in a parametrization can be exploited to achieve a significant reduction in the LP formulation, which can lead to polynomial complexity of such realization problems - some parametrizations even resulting in the constraint polytope being independent of d. We also explore the use of a probabilistic viewpoint on TDMs to derive the constraint polytopes.

中文翻译：

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关于尾部依赖矩阵

在双变量尾部依赖度量中，尾部依赖系数已成为流行的选择。类似于相关矩阵，使用这些双变量测度构造了一个多元依赖测度，在文献中将其称为尾部依赖矩阵（TDM）。虽然确定给定d × d矩阵是否为相关矩阵的问题的阶数为O（d ³在复杂性方面，确定矩阵是否为TDM（实现问题）被认为是非多项式的。使用线性规划（LP）公式，我们表明约束的组合结构与加权图中的难处理的最大割问题有关。该连接为构造允许在d中获得多项式的参数类提供了途径确定其约束多面体成员的算法。一般实现问题的复杂性理所当然地在理论上引起了很大兴趣。由于在实践中，人们诉诸于TDM的低维参数化，因此我们认为，实际问题是，限于TDM的参数类别的实现问题的复杂性及其算法。在本文中，我们展示了如何利用参数化中固有的对称性和稀疏性来实现LP公式的显着减少，这可能导致此类实现问题的多项式复杂性-一些参数化甚至导致约束多态性独立于d。我们还探索了在TDM上使用概率观点来导出约束多表位的方法。