二元COPULA擾動構造法的研究
發(fā)布時間：2018-08-26 18:49
【摘要】：Copula函數可以捕捉到變量之間非線性、非對稱以及分布尾部的相關關系,是解決其他學科相關問題的重要工具,特別在研究兩個變量的相關關系中有重大的應用.Copula的構造理論是Copula理論的基本問題之一,也是進行其實際問題應用的理論基礎.隨著Copula在社會各領域的廣泛應用,一些已有的Copula可能不能滿足解決問題的需要,通過構造新型的Copula,對原有的Copula進行改進創(chuàng)新,可以得到原有Copula不具備的某些性質,找到更適合解決實際問題的 Copula.本文研究的是二元Copula的擾動構造法,其基本思想是通過對原有Copula添加適當的擾動項來構造新的二元Copula.其中也可以得到著名的FGM-Copula族和Plackeet-Copula族,但是卻不同于以往的代數構造法,這種方法更為靈活,也更為簡單,對Copula的包容性也更強,這也是擾動項構造法的意義所在.首先,本文研究了乘積Copula的擾動構造問題.乘積Copula描述的是隨機變量X,Y間的獨立性,然而實際情況往往是隨機變量X,Y并不是相互獨立的,而對乘積Copula添加符合條件的擾動項就可以打破對變量獨立性要求的限制,進而可以研究變量間的相依性,更具有現實意義.乘積Copula擾動構造的具體形式為Cλ(u,v)=uv+λf(u)g(v),其中λf(u)g(v)即為擾動項,文中給出了使得Cλ(u,1v)為Copula的充要條件并研究了該Copula族的相依性等性質.在此基礎上繼而給出了乘積Copula擾動構造的高次冪拓展和多參數拓展,形式分別為Cλ(u,v)= uv+λavb(1-u)c(1-v)d,Cλ(u,v)=uv+(?)λifi(u)gi(v),在這兩種形式下可以得到廣義的FGM-Copula族,從而實現了對FGM-Copula族的拓展.相對于FGM-Copula的和諧性度量,新型擾動Copula的和諧性度量的取值范圍更廣,體現了其在研究變量間的相依性方面的優(yōu)良性質.更進一步地,本文討論了一般二元Copula的擾動構造,其具體形式為CNλ=C+λ(u-C)(v-C),其中Nλ=λ(u-C)(v-C)便為其擾動項.同樣地,在此方法的基礎上做線性凸組合進行復合擾動構造,得到雙參數Copula CNα,β,其中CNα,β= αC+ β +(1-α-β)C(u+v-c).本文后面討論了CNα,β與Plackeet-Copula族的聯(lián)系,以及其在次序和,不變性,Schur-凹性方面的若干性質.
[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 偽, 尾, where CN 偽, 尾 = 偽 C 尾 (1- 偽-尾) C (u v-c) is obtained. In this paper, we discuss the relation between CN 偽, 尾 and Plackeet-Copula family, and some properties of CN 偽, 尾 in order sum, invariance, Schur-concave property.
【學位授予單位】：天津工業(yè)大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：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 偽, 尾, where CN 偽, 尾 = 偽 C 尾 (1- 偽-尾) C (u v-c) is obtained. In this paper, we discuss the relation between CN 偽, 尾 and Plackeet-Copula family, and some properties of CN 偽, 尾 in order sum, invariance, Schur-concave property.
【學位授予單位】：天津工業(yè)大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：F224
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