二元Archimax Copulas的可识别性、收敛性和非参数估计
Identifiability, Convergence and Nonparametric Estimation of Bivariate Archimax Copulas
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中文总结 AI 辅助
研究二元Archimax Copulas的可识别性等问题，通过变换生成元和相依函数解决可识别性，利用收敛等价关系构造非参数估计量，证明其强一致性，模拟研究表明CFG型估计量性能优，还给出实际数据应用。
中文摘要 AI 辅助
考虑到二元Archimax Copulas族既包含阿基米德型又包含极值型，Archimax Copulas构成了一个灵活的族，可用于对极端和中等程度的相依性进行建模。尽管它们很有吸引力，但尚未建立一个完全非参数、一致的估计量，且该估计量本身是Archimax族$\mathcal{C}_{am}$的一个元素，主要是因为仅阿基米德生成元和Pickands相依函数不能识别Archimax Copulas。我们通过使用变换后的生成元和变换后的Pickands相依函数解决了这个可识别性问题，并表明这些函数确实能唯一识别Archimax Copula。基于此结果，我们证明了Archimax Copulas的一致收敛等价于相应变换后的生成元和Pickands相依函数的一致收敛。此外，对于阿基米德型和极值型Copulas，$\mathcal{C}_{am}$中的一致收敛等价于几乎所有条件分布的弱收敛。利用这些等价关系，我们构造了两个用于Archimax Copulas的非参数估计量（一个Pickands型和一个CFG型估计量，均为$\mathcal{C}_{am}$的元素），并表明它们在温和的正则条件下是强一致的。作为上述弱条件收敛的进一步结果，我们得到了用于定向相依性度量的强一致插件估计量，如Chatterjee的$\xi$和Trutschnig的$\zeta_1$。大规模模拟研究表明，所提出的CFG型估计量优于标准经验Copula估计量和Pickands型估计量；对奥地利布雷根茨和多恩比恩降水数据的应用说明了我们的估计量在实际数据上的实际应用。
英文摘要
Considering that the family of bivariate Archimax copulas contains both the Archimedean and the extreme-value class, Archimax copulas constitute a flexible family allowing to model extreme and moderate levels of dependence. Despite their appeal, no fully nonparametric, consistent estimator that is itself an element of the Archimax family $\mathcal{C}_{am}$ has been established yet, mainly because Archimedean generators and Pickands dependence functions alone do not identify Archimax copulas. We resolve this identifiability issue by working with transformed generators and transformed Pickands dependence functions, and show that these functions do identify the Archimax copula uniquely. Building upon this result, we prove that uniform convergence of Archimax copulas is equivalent to uniform convergence of the corresponding transformed generators and Pickands dependence functions. Moreover, as for Archimedean and extreme-value copulas, uniform convergence in $\mathcal{C}_{am}$ is equivalent to weak convergence of almost all conditional distributions. Exploiting these equivalences, we construct two nonparametric estimators for Archimax copulas (a Pickands and a CFG type estimator, both elements of $\mathcal{C}_{am}$) and show that they are strongly consistent under mild regularity conditions. As a further consequence of the aforementioned weak conditional convergence, we obtain strongly consistent plug-in estimators for measures of directed dependence such as Chatterjee's $ξ$ and Trutschnig's $ζ_1$. A large-scale simulation study shows that the proposed CFG type estimator outperforms both the standard empirical copula estimator and the Pickands type estimator; an application to precipitation data from Bregenz and Dornbirn (Austria) illustrates the practical use of our estimators on real data.
