Conditional Archimedean COPULA

Dependence relations between random variables is one of the most widely studiedsubjects in probability and statistics.The theory of copula is more and more perfect,and is used more and more widely. We may have know the dependence relation of some variables,but how to use the known dependence relation to detect dependent structure of the unkown variables or the effect on the unkown variables,and how the known variables have an effect on the unkown variables and so on,which are all our concerned questions.So in this article ,we mainly talk about sevarel types of dependence relation between the conditional distribution functions and the corresponding conditional copulas, conditional Copulas and unconditional copulas,which are variable and invariable,and how they vary.In this article,we mainly discuss the properties of conditional Copulas,and get some important conclusions.In the second chapter,we get this conclusion:Theconditional Archimedean Copulas C_[a,b](u,v) are still Archimedean copulas andthe inverse generater isφ_[a,b]^-1(t)=φ^-1(t+φ(b))-φ^-1(t+φ(a))/b-a.We alsoconsider the conditional. Archimedean Copulas and their properties under the condition W≤w,including (1)orderness:when w≤z ,C_[0,w](u,v)≤C_[0,z](u,v) andconditional Copulas preserves the concordance relation between two ArchimedeanCopulas;(2)boundness: conditional Copulas preserves the Frecher upper bound;(3)convergrnce: whenφ∈R_-α(0<α<∞),lim(?) C_[0,w](u,v)=C^C1,α(u,v)ï¼›(4)tail dependence: if (?) a≥0 and lim(?)φ^-1(x)/φ^-1(x+a)=1,the conditional lower tailindex of C_[0,w](u,v)λ_[0,w]equals lower tail index of Cλ。In the third chapter,,we get the following conclusion:the upper bound can be improved to M₁ (u, v) =uv/u+v-uvand some other usefull conclusion.In the fourth chapter,we consider the properties of conditional copulas under the condition min(X,Y)>t and max(X,Y)≤t ,wealso find the similarity beteen the Copulas under these conditions and the conditional Archimedean copulas in the second chapter.