Research On Binary COPULA Disturbance Construction Method

The Copula function can capture the correlation between variables of nonlinear,asymmetric and tail distribution.It is an important tool to solve problems related to other disciplines,especially in the study of correlation between the two variables in the structural theory of application.The construction theory of Copula is one of the basic problems of Copula theory,and it is also the theoretical basis of its practical application.As Copula is widely used in various fields of society,some of the Copula may not be able to meet the needs of solving problems,by constructing a new type of Copula,to make some difference in the original Copula,can get some properties of the original Copula that does not have,and to find more suitable Copula to solve the practical problems.In this paper,we study the perturbation method of the bivariate Copula.The basic idea is to construct a new bivariate Copula by adding an appropriate perturbation term to the original Copula,which can also get the famous FGM-Copula and Plackeet-Copula family,but it is different from the previous algebraic construction method,this method is more flexible,and more simple,more inclusive of Copula,which is also the significance of the construction method of perturbation.First,this paper studies the perturbation structure of product Copula.Product Copula describes the independence between the random variables X,Y,however,the actual situation is often that the random variables are not independent of each other,adding the perturbation term to the product can break the limitation of the requirement of the independence of variables.Then we can study the dependence between variables,but also has practical significance.The specific form of the perturbation structure of product Copula is:C?(u,v)=uv+?f(u)g(v),which ?f(u)g(v)is the disturbance term,the necessary and sufficient conditions for Copula C?(u,v)are given and the properties of the Copula family are studied.On the basis of the above,we give the power expansion and multi parameter extension of product Copula perturbation structure,they are in the form of C?(u,v)= uv + ?vavb(1-u)c(1-v)dand C?(u,v)=uv+(?)?ifi(u)gi(v),in these two forms,we can get the generalized FGM-Copula family,so we can extend the FGM-Copula family.Compared with the concordance measure of FGM-Copula,the new type of disturbance Copula has a wider range of values,which reflects the excellent nature of the dependence between variables in the study.Furthermore,this paper discusses the general perturbation structure of the bivariate Copula,the specific form of the Copula is:CN?=C + ?(u-C)(v-C),which N?= ?(u-C)(v-C)is the perturbation term.Similarly,a linear convex combination is constructed based on this method,two parameters Copula CN?,? can be obtained under this method,like that CN?,?q=?C+?uv+(1-?-?)C(u+v-c).In the later of this paper,we discuss the relations between CN?,? and Plackeet-Copula,as well as some properties such like ordinal sums,invariance,Schur-concave and so on.