极值Copula的构造及其性质的研究

发布时间：2019-01-22 16:05

【摘要】：极值统计是研究极端事件的有效方法,它广泛应用于金融、保险、水文、环境等领域,由于现实世界的复杂性,极值事件越来越倾向于同时或相继发生,比如金融传染、投资组合风险管理、多地区大气污染、极端天气的日益频繁等等,因此多元极值统计正成为极值统计学的理论前沿和研究热点. 多元极值统计的理论研究中,极值Copula是不可或缺的分析工具,它用于刻画极端事件之间的相关结构,以及多元极值分布的模型表示.构造多元极值分布是多元极值统计中研究的一个基本问题,它等价于构造极值Copula.本文主要通过Copula的乘积构造极值Copula,首先给出了构造模型,这个模型由一些已知的极值Copula和单位区间上的一系列函数构成,由于极值Copula具有最大恒稳定性,模型中函数的形式也限定为一元幂函数.其次,讨论了这类新极值Copula的一个边界性质,以及特殊情况下新极值Copula的模型解释,结论表明,k个Copula的标准凸组合所属吸引场可用它们各自吸引场的加权几何平均表示. 尾部相关系数用于度量多元随机变量的尾部一致性,文中给出了多元极值Cop-ula的上象限尾部相关系数表达式,并得到了新的极值Copula二元尾部相关系数,以及多元上象限尾部相关系数的解析表达式.最后,本文以Gumbel Copula分别与最大Copula和乘积Copula组合为例,对得到的两个新极值Copula进行分析,讨论了它们的尾部相关性及其相关结构的非对称性.
[Abstract]:Extreme value statistics is an effective method for studying extreme events. It is widely used in the fields of finance, insurance, hydrology, environment and so on. Due to the complexity of the real world, extreme events tend to occur simultaneously or sequentially, such as financial contagion. Portfolio risk management, multi-regional air pollution, the increasing frequency of extreme weather, and so on, so multivariate extreme statistics is becoming the theoretical frontier and research hotspot of extreme value statistics. In the theoretical study of multivariate extremum statistics, extreme Copula is an indispensable analytical tool, which is used to describe the correlation structure between extreme events and the model representation of multivariate extreme distribution. Constructing multivariate extremum distribution is a basic problem in multivariate extreme value statistics, which is equivalent to constructing extreme value Copula.. In this paper, we first give the construction model of extreme value Copula, by the product of Copula. This model is composed of some known extreme value Copula and a series of functions on the unit interval, because the extreme value Copula has the maximum constant stability. The form of function in the model is also limited to a univariate power function. Secondly, a boundary property of this kind of new extreme Copula is discussed, and the model interpretation of the new extreme value Copula in special cases is discussed. It is shown that the standard convex combinations of k Copula can be represented by the weighted geometric average of their respective attraction fields. The tail correlation coefficient is used to measure the tail consistency of multivariate random variables. In this paper, the upper quadrant tail correlation coefficient expression of multivariate extreme value Cop-ula is given, and a new extreme Copula binary tail correlation coefficient is obtained. And the analytical expression of tail correlation coefficient in multivariate upper quadrant. Finally, taking the combination of Gumbel Copula, maximum Copula and product Copula as examples, the two new extremum Copula obtained are analyzed, and the tail correlation and asymmetry of their related structures are discussed.
【学位授予单位】：天津大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：F224

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