Two New Dependence Stochastic Orders Based On Copulas

Stochastic order is very important in statistics and probability. And many stochastic orders are applied to many fields, such as Finance, Insurance and Actuary, Reliability Theory, Survival Analysis, Communication System. Following applications with stochastic orders and copulas, we can use stochastic orders together with copulas. So we must study dependence properties of stochastic orders. In this paper, we present concepts of upper orthant hazard order and lower orthant quasi hazard order and study basic properties and applications of them.In first chapter, we introduce some basic stochastic orders and the definition of copula. And we also give some properties of stochastic orders and copulas.Let X = (X₁,X₂,…, X_n) and Y = (Y₁, Y₂,…, Y_n) be two random vectors with respective survival functions F and G. we say that X is smaller than Y according to hazard rate order(denoted by X≤hr Y or F≤hr G) if (G|—)(x)/(F|—)(x) is increasing in x∈{x : (G|—)(x) > 0}.Similarly, there is the definition of quasi hazard order. we say that X is smaller than Y according to quasi hazard rate order(denoted by X≤qh Y or F≤qh G) if (G|—)(x)/(F|—)(x) is decreasing in x∈{x : G(x) > 0}.Next there is an important theorem of Sklar for copulas.Sklar theorem Let F be an n-dimensional distribution function with continuous margins F₁,…,F_n. Then F has a unique copula representation: F(x-1,…,x_n)=C(F₁(x₁),…,F_n(x_n)).We also introduce the definitions of K endall'Ï„and Spearman'ρof Archimedean copulas.In second chapter, we introduce the definitions of upper orthant hazard order and lower orthant quasi hazard order. And we study some properties of them, relationship with other stochastic orders. We consider properties and applications of this two stochastic orders with copulas. At last we give many examples.In this chapter, we introduce the following definitions and properties.Let X = (X₁,X₂,…,X_n) and Y = (Y₁, Y₂,…,Y_n) be two random vectors with respective survival functions (F|—) and (G|—).Definition 1 We say that X is smaller than Y according to upper orthant hazard rate order(denoted by X =uohr Y or F=uohr G) if for all u_j∈[0,1], i, j = 1,…n, j≠i. We use the convention a/0≡∞, for a > 0.Similarly, we can define the lower orthant quasi hazard rate order.Definition 2 We say that X is smaller than Y according to lower orthant quasi hazard rate order(denoted by X≤lorh Y or F≤lorh G) if for all u_j∈[0,1], i, j = 1,…,n,j≠i. We use the convention a/0≡∞, for a > 0.Theorem 3 Let X and Y be two n-dimensional random vectors, if X≤uohr Y(X≤loqh Y), then where I = {i₁,i₂,…,i_k} (?) {1,2,…,n}, X_I = (X_i1,…,X_ik), Y_I = {Y_i1,…,Y_ik), k = 1,…,n. Theorem 4 If F∈F₂(F_i F_j), X = (X_i, X_j), i≠j, then F^-≤uohr F,F^-≤loqh F.Theorem 5 Let X and Y have, respectively, continuous distribution functions F and G in F_n(F₁, F₂,…, F_n), then X≤uohr Y(X≤loqh Y)if and only if there exist R_F and Rg such that R_F≤uohr R_G(R_F≤loqh R_G).Theorem 6 1.If X_α≤uohr X_β, for allα≤β, if the partial derivatives below exists , if and only if2.If X_α≤loqh X_β, for allα≤β, if the partial derivatives below exists, if and only ifTheorem 7 Let X and Y be two n-dimensional random vectors which have the same copula, then X≤uohr Y (?) X_i≤uo Y_i = 1,…,n,X≤logh Y (?) X_i≥lo Y_i = 1,…,n.In third chapter, firstly, we introduce the method of estimate Archimedean copulas. Using to estimate the parameter of Archimedean copulas. Secondly, we introduce that how to find the goodness-of-fit of Archimedean copulas, and how to estimate Archimedean copulas whose variables are Shenzhen stock of A and Shenzhen stock of B, Shanghai sotck of A and Shanghai stock of B, respectively, through the index of stock market of Shenzhen and Shanghai. We test that the distribution of the index of Shenzhen stock of A under the condition of the index of Shenzhen stock of B is less than the distribution of the index of Shanghai stock of A under the condition of the index of Shanghai stock of B in lower orthant quasi hazard order. It satisfies R_α≤loqh R_β.There is that the ratio of the distribution of the index of stock market of Shanghai and the distribution of the index of stock market of Shenzhen will diminish gradually when indexes are increasing.