摘要
随机变量之间的相依性,是概率论与数理统计研究的核心。随机变量的联合分布完全决定了其各个分量的分布以及相依关系。由于多维随机变量的联合分布比较难确定,而其边缘分布是可以得到的,将边缘分布与相依关系区分开来分别进行研究是必要的。
一个多维分布的连接函数是一种描述其各个分量间怎样相互依赖的函数,它不考虑边缘分布的影响,而只聚焦于相依结构。Copula就是这样一种连接函数,近年来对Copula的研究已有很多。Copula已经很广泛地应用到了经济领域和随机过程方面。
本文主要通过Copula的加权几何平均来构造Copula的。构造Copula的方法已经有很多了,有通过生成元、已知Copula的偏导或其线性组合来构造的,还有通过中间函数g函数来构造的。本文是通过已知Copula的加权几何平均来构造Copula。
文中给出了一类新的Copula——WGM Copula,它是正象限相依的。用其加权几何平均来构造的Copula仍为WGM Copula。Cuadras-Auge Copula就是用WGM Copula M=min(u,v)和Π=uv的加权几何平均来构造的,它具有很好的上尾相依性。本文用Cuadras-Auge Copula和Gumbel Copula对数据进行刻画并比较其优劣,得出Cuadras-Auge Copula稍优的结论。
Dependence relation between random variables is the main subject in Probability and Statistics. The joint distribution of random variables can totally tell us the distribution of its components and the dependence relation between them. For multivariate random variables, it is difficult to fix on their joint distribution, though we can easily get its marginal distributions. It is essential to study the marginal distributions and the dependence relation between them dividually.
The link function of multivariate random variables can characterize the function relation between them, without considering about the influence of its margins and only focus on its dependence structure. Copula is a kind of such link function, and it has been studied widely. Copula as so far has been applied to fields as finance and stochastic process.
The main subject of the paper is to construct bivariate Copula. Copula has been constructed through a generator or a indirect function g or by making use of known Copulas with their deviations or the weighted arithmetic mean of them. In the paper, it is concerned about the weighted geometric mean of known Copulas.
There is given a new kind of Copula—WGM Copula which has the property of positively quadrant dependence. The weighted geometricmean of WGM Copulas M = min(u,v) andΠ=uv such as Cuadras-Auge
Copula is still WGM Copula. Cuadras-Auge Copula is has better property of upper tail dependence. Comparing Cuadras-Auge Copula with Gumbel Copula in characterizing practical data, we find that Cuadras-Auge Copula is better in analyzing data with upper tail dependence.
引文
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