二元COPULA扰动构造法的研究
发布时间：2018-08-26 18:49
【摘要】：Copula函数可以捕捉到变量之间非线性、非对称以及分布尾部的相关关系,是解决其他学科相关问题的重要工具,特别在研究两个变量的相关关系中有重大的应用.Copula的构造理论是Copula理论的基本问题之一,也是进行其实际问题应用的理论基础.随着Copula在社会各领域的广泛应用,一些已有的Copula可能不能满足解决问题的需要,通过构造新型的Copula,对原有的Copula进行改进创新,可以得到原有Copula不具备的某些性质,找到更适合解决实际问题的 Copula.本文研究的是二元Copula的扰动构造法,其基本思想是通过对原有Copula添加适当的扰动项来构造新的二元Copula.其中也可以得到著名的FGM-Copula族和Plackeet-Copula族,但是却不同于以往的代数构造法,这种方法更为灵活,也更为简单,对Copula的包容性也更强,这也是扰动项构造法的意义所在.首先,本文研究了乘积Copula的扰动构造问题.乘积Copula描述的是随机变量X,Y间的独立性,然而实际情况往往是随机变量X,Y并不是相互独立的,而对乘积Copula添加符合条件的扰动项就可以打破对变量独立性要求的限制,进而可以研究变量间的相依性,更具有现实意义.乘积Copula扰动构造的具体形式为Cλ(u,v)=uv+λf(u)g(v),其中λf(u)g(v)即为扰动项,文中给出了使得Cλ(u,1v)为Copula的充要条件并研究了该Copula族的相依性等性质.在此基础上继而给出了乘积Copula扰动构造的高次幂拓展和多参数拓展,形式分别为Cλ(u,v)= uv+λavb(1-u)c(1-v)d,Cλ(u,v)=uv+(?)λifi(u)gi(v),在这两种形式下可以得到广义的FGM-Copula族,从而实现了对FGM-Copula族的拓展.相对于FGM-Copula的和谐性度量,新型扰动Copula的和谐性度量的取值范围更广,体现了其在研究变量间的相依性方面的优良性质.更进一步地,本文讨论了一般二元Copula的扰动构造,其具体形式为CNλ=C+λ(u-C)(v-C),其中Nλ=λ(u-C)(v-C)便为其扰动项.同样地,在此方法的基础上做线性凸组合进行复合扰动构造,得到双参数Copula CNα,β,其中CNα,β= αC+ β +(1-α-β)C(u+v-c).本文后面讨论了CNα,β与Plackeet-Copula族的联系,以及其在次序和,不变性,Schur-凹性方面的若干性质.
[Abstract]:The Copula function can capture the nonlinear, asymmetric and distributed tail correlation between variables, and is an important tool to solve related problems in other disciplines. Especially in the study of the correlation between two variables, the construction theory of .Copula is one of the basic problems of Copula theory, and also the theoretical basis of its practical application. With the wide application of Copula in various fields of society, some existing Copula may not be able to meet the needs of solving the problem. By constructing a new type of Copula, to improve and innovate the original Copula, some properties that the original Copula does not possess can be obtained. Finding a Copula. that is better suited to solving practical problems In this paper, the perturbation construction method of binary Copula is studied. Its basic idea is to construct a new binary Copula. by adding appropriate perturbation terms to the original Copula. The famous FGM-Copula family and Plackeet-Copula family can also be obtained, but different from the previous algebraic construction method, this method is more flexible, simpler and more inclusive to Copula, which is the meaning of the perturbation term construction method. Firstly, the perturbed construction of product Copula is studied. The product Copula describes the independence of the random variable XY, but the actual situation is that the random variable XY is not independent of each other. However, adding a qualified perturbation term to the product Copula can break the restriction on the independence of the variable. Furthermore, it is of practical significance to study the dependence of variables. The concrete form of product Copula perturbation construction is C 位 (UV) uv 位 f (u) g (v), where 位 f (u) g (v) is the perturbation term. In this paper, the necessary and sufficient conditions for C 位 (UH 1v) to be Copula are given, and the dependence of the Copula family is studied. On this basis, the higher power extension and the multiparameter extension of the product Copula perturbation construction are given, respectively, in the form of C 位 (UV) = uv 位 avb (1-u) c (1-v) DU C 位 (UV) UV (?) 位 ifi (u) gi (v),. In these two forms, the generalized FGM-Copula family can be obtained, thus the extension of FGM-Copula family can be realized in the form of C 位 (UV) = uv 位 avb (1-u) c (1-v) DU C 位 (UV) UV (?) 位 ifi (u) gi (v),. Compared with the harmoniousness measurement of FGM-Copula, the new perturbed Copula has a wider range of harmoniousness measures, which reflects its excellent properties in studying the dependence of variables. Furthermore, the perturbation construction of a general binary Copula is discussed in this paper. Its concrete form is CN 位 C 位 (u-C) (v-C), where N 位 = 位 (u-C) (v-C) is its perturbation term. Similarly, on the basis of this method, the compound perturbation is constructed by linear convex combination, and the two-parameter Copula CN 伪, 尾, where CN 伪, 尾 = 伪 C 尾 (1- 伪-尾) C (u v-c) is obtained. In this paper, we discuss the relation between CN 伪, 尾 and Plackeet-Copula family, and some properties of CN 伪, 尾 in order sum, invariance, Schur-concave property.
【学位授予单位】：天津工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：F224
[Abstract]:The Copula function can capture the nonlinear, asymmetric and distributed tail correlation between variables, and is an important tool to solve related problems in other disciplines. Especially in the study of the correlation between two variables, the construction theory of .Copula is one of the basic problems of Copula theory, and also the theoretical basis of its practical application. With the wide application of Copula in various fields of society, some existing Copula may not be able to meet the needs of solving the problem. By constructing a new type of Copula, to improve and innovate the original Copula, some properties that the original Copula does not possess can be obtained. Finding a Copula. that is better suited to solving practical problems In this paper, the perturbation construction method of binary Copula is studied. Its basic idea is to construct a new binary Copula. by adding appropriate perturbation terms to the original Copula. The famous FGM-Copula family and Plackeet-Copula family can also be obtained, but different from the previous algebraic construction method, this method is more flexible, simpler and more inclusive to Copula, which is the meaning of the perturbation term construction method. Firstly, the perturbed construction of product Copula is studied. The product Copula describes the independence of the random variable XY, but the actual situation is that the random variable XY is not independent of each other. However, adding a qualified perturbation term to the product Copula can break the restriction on the independence of the variable. Furthermore, it is of practical significance to study the dependence of variables. The concrete form of product Copula perturbation construction is C 位 (UV) uv 位 f (u) g (v), where 位 f (u) g (v) is the perturbation term. In this paper, the necessary and sufficient conditions for C 位 (UH 1v) to be Copula are given, and the dependence of the Copula family is studied. On this basis, the higher power extension and the multiparameter extension of the product Copula perturbation construction are given, respectively, in the form of C 位 (UV) = uv 位 avb (1-u) c (1-v) DU C 位 (UV) UV (?) 位 ifi (u) gi (v),. In these two forms, the generalized FGM-Copula family can be obtained, thus the extension of FGM-Copula family can be realized in the form of C 位 (UV) = uv 位 avb (1-u) c (1-v) DU C 位 (UV) UV (?) 位 ifi (u) gi (v),. Compared with the harmoniousness measurement of FGM-Copula, the new perturbed Copula has a wider range of harmoniousness measures, which reflects its excellent properties in studying the dependence of variables. Furthermore, the perturbation construction of a general binary Copula is discussed in this paper. Its concrete form is CN 位 C 位 (u-C) (v-C), where N 位 = 位 (u-C) (v-C) is its perturbation term. Similarly, on the basis of this method, the compound perturbation is constructed by linear convex combination, and the two-parameter Copula CN 伪, 尾, where CN 伪, 尾 = 伪 C 尾 (1- 伪-尾) C (u v-c) is obtained. In this paper, we discuss the relation between CN 伪, 尾 and Plackeet-Copula family, and some properties of CN 伪, 尾 in order sum, invariance, Schur-concave property.
【学位授予单位】：天津工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：F224
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