A large number of commonly used parametric Archimedean copula (AC) families are restricted to a single parameter, connected to a concordance measure such as Kendall's tau. This often leads to poor statistical fits, particularly in the joint tails, and can sometimes even limit the ability to model concordance or tail dependence mathematically. This work suggests outer power (OP) transformations of Archimedean generators to overcome these limitations. The copulas generated by OP-transformed generators can, for example, allow one to capture both a given concordance measure and a tail dependence coefficient simultaneously. For exchangeable OP-transformed ACs, a formula for computing tail dependence coefficients is obtained, as well as two feasible OP AC estimators are proposed and their properties studied by simulation. For hierarchical extensions of OP-transformed ACs, a new construction principle, efficient sampling and parameter estimation are addressed. By simulation, convergence rate and standard errors of the proposed estimator are studied. Excellent tail fitting capabilities of OP-transformed hierarchical AC models are demonstrated in a risk management application. The results show that the OP transformation is able to improve the statistical fit of exchangeable ACs, particularly of those that cannot capture upper tail dependence or strong concordance, as well as the statistical fit of hierarchical ACs, especially in terms of tail dependence and higher dimensions. Given how comparably simple it is to include OP transformations into existing exchangeable and hierarchical AC models, this transformation provides an attractive trade-off between computational effort and statistical improvement.

中文翻译：

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分层阿基米德联结的外幂变换：构造、采样和估计

大量常用的参数阿基米德联结 (AC) 族仅限于单个参数，连接到诸如 Kendall tau 之类的一致性度量。这通常会导致统计拟合不佳，尤其是在联合尾部，有时甚至会限制以数学方式对一致性或尾部相关性进行建模的能力。这项工作建议使用阿基米德生成器的外功率 (OP) 变换来克服这些限制。例如，由 OP 转换生成器生成的 copula 可以允许同时捕获给定的一致性度量和尾部相关系数。对于可交换的OP变换AC，获得了尾部相关系数的计算公式，并提出了两个可行的OP AC估计量，并通过仿真研究了它们的性质。对于 OP 转换的 AC 的分层扩展，解决了新的构造原理、有效采样和参数估计。通过仿真，研究了所提出的估计器的收敛速度和标准误差。在风险管理应用程序中展示了 OP 转换的分层 AC 模型的出色尾部拟合能力。结果表明，OP 变换能够改善可交换 ACs 的统计拟合，特别是那些不能捕获上尾依赖或强一致性的 ACs，以及分层 ACs 的统计拟合，特别是在尾依赖和更高维度方面. 鉴于将 OP 转换包含到现有的可交换和分层 AC 模型中是多么简单，