On Densities of Extreme Value Copulas
On Densities of Extreme Value Copulas
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发表时间：
2013
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通讯作者：
M. Sc;Thesis;Gabrielle Doyon;M. Hofert
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作者：
M. Sc;Thesis;Gabrielle Doyon;M. Hofert
在文献中可以找到关于极值连体（EVCs）的大量工作，但它们的相关密度主要是在二元情况下研究的。这项工作旨在弥合二元和多元密度之间的差距，因为密度的显式表达在许多多元统计应用中是最重要的。在简要概述了相关理论之后，我们提出了EVCs密度的一般公式。该公式依赖于联结的底层稳定尾相关函数的导数，因此我们简要回顾Smith分布和多元t分布的稳定尾相关函数的构造原理，参见Joe et al.（2008）。Genest和Rivest（1989）给出了一个基于阿基米德copulas的结果，该结果具有在1处有规则变化且尾指数大于1的逆生成器，并且基于它们对稳定尾相关函数的表示，我们可以将我们的公式应用于一般EVC的密度，并从阿基米德copulas构造新的可处理EVC密度。在类似的假设条件下，我们可以将构造推广到嵌套的阿基米德copulas，并可以推导出嵌套阿基米德copulas的稳定尾依赖函数。最后，我们得到了N个最大阶统计量的联结密度的隐式表达式。
Extensive work on extreme value copulas (EVCs) can be found in the literature, but their associated densities are mainly studied in the bivariate case. This work aims to bridge the gap between bivariate and multivariate densities, as an explicit expression of the density is of utmost importance in many multivariate statistical applications. After a brief overview of copula-related theory, we present a general formula for the density of EVCs. This formula depends on the derivatives of the underlying stable tail dependence function of the copula, so we briefly recall the construction principle of the stable tail dependence function of the Smith and multivariate t distributions, see also Joe et al. (2008). A result based on Archimedean copulas with inverse generators that are regularly varying at one with tail index bigger than one is presented in Genest and Rivest (1989), and based on their representation of the stable tail dependence function, we can apply our formula for the density of a general EVC and construct new tractable EVC densities from Archimedean copulas. Under similar assumptions, we can extend the construction to nested Archimedean copulas and we are able to derive the stable tail dependence function of nested Archimedean copulas. Finally, we obtain an implicit expression for the density of the copula of the N -largest order-statistics.