Bivariate COPULAs With Polynomial Sections

The study of copulas and their applications in statistics is a rather modern phenomenon. Why are copulas of interest in probability and statistics? Answers from a book of Nelsen (1999), two main reasons: Firstly, as a way of studying scale-free measures of dependence, and secondly, as a starting point for constructing families of bivariate distributions , sometimes with a view to simulation. Although great development has been made in the past decades, the study of copulas and role they play in probability, statistics, stochastic processes is a subject still in its infancy. There are many open problems and much work to be done.In this paper, we will deal with an open problem left by Nelsen (1997). In section 4 of chapter 3,we generalize a method used by Nelsen et. al. [4] to generate a wide range of families of bivariate copulas by a generalized Bernstein polynomial formula. The method requires that the intersection of the copula in the plane will be a polynomial with respect to x. Obviously y and x can be interchanged. We present necessary and sufficient conditions for bivariate copulas with 4-degree polynomial sections and sufficient conditions for higher order. We obtain the necessary and sufficient conditions that random vector with associated copula with polynomial sections is radially symmetric or jointly symmetric. In chapter 4, we derive general formulas of Spearman's ρ and Kendall's τ for copulas with polynomial sections and correct the upper bound of rho on copulaswith cubic sections. Examples of symmetric copulas with 4-degree polynomial sections are given to demonstrate how one can construct copulas with desired positive dependence properties.There is a interesting interpretation on the sections of copulas. When X and Y are uniform (0,1) random variables with joint distribution function C, the sections are proportional to conditional distribution functions. That is, for yo € /,Pr{X < x\Y < y0} = Pr{X 2, ao(y), ai(y),..., an(y) are appropriate functions onBernstein models from rewritting basic model .?C(x, y) = xy + My)z(l - x)""1 + ? ? ? + b^^x^1 (1 - x) (1)where x,y e I,n > 2,h(y),b2{y),.. .,6n_i(y) are appropriate functions on /.Main conclusions of this paper:Theorem 3.14 Let bi,i = l,...,n— 1 : / —>? R, be functions satisfying 6^(0) = 6?(1) = 0,i = l,...n - 1, n > 2, and let C{x,y) be function defined by (1), Then C is a Copula if and only if for every xi,X2,yi,y2 e I such that x\ < x2,yi < yi, we haver1] ■ &-1)S (" 1k) T **"* >-Theorem 3.15 Let bi(y),i = l,...,n- l,C(x,y) be as Theorem 3.14 , Then if the next two conditions are satisfied:i) bi(y)>i = l,..-,n— 1 are absolutely continuous and. ii) for all x E I and almost every y e / ,^ [(** - k)xk-\x -1)"-*-1] > oThen , C is a Copula. Conversely , if bi(y),i = l,...,n — 1 are differentiable on 7 , ii) holds .Theorem 3.16 Let bi(y),i = l,...,n - l,C(x,y) be as Theorem 3.14 , Theni) (X,Y) with associated copula C is radially symmetric if and only ify),i l,...,l,n 2k;bi{y) = &n-i(l - y),i = 1, ? ? ?, ^-,n = 2k + 1;ii) (X,Y) with associated copula C is jointly symmetric if and only ifbi(y) = -bi(l -y) = -bn-i(y) = 6^(1 - y), 6f (y) = 0,t = l,...>^-l,n = 2fc;bi(y) = -bi(l - y) = -6n_i(j/) = 6n_i(l - y),Theorem 3.17 Let 6,,i = 1,2,3 : / —> R,be functions satisfying 6j(0) = bi(l) = 0,i = 1,2,3, and let C{x,y) = xy+bl(y)x{l-x)3+b2(y)x2{l-x)2 + &3(y)x3(l — x), Then C is a Copula if and only ifi) bi(y),i = 1,2,3 are absolutely continuous.ii) for all x 6 / and for almost every y € / ,(4*4] [ k)xk-\x - i)3-fc] > oMoreover, C are absolutely continuous.Theorem 4.1 For the copula C defined by (1), the measure of dependence p and r are :n-lpc = 12^2Beta(i + l,n-i + l) f bi(y)dy ?=i J°n—In—1= 4 E Y t=i j=in-lwhere , Beta(i,j) = r(i)T(j)/r(i + j), T(i) = (i- 1)!.Remark: The precise bound of Spearman's p on bivariate copulas with cubic sections is 0.68125. whereC(x,y) = xy + bi(y)x(l - x)2 + b2(y)x2(l -x), , x . 2.245829494y 0 < y < 0.3033b\\y) = <1 0.977880301(1 - y) 0.3033 < y < 10.977880301y 0 < y < 0.30332.245829494(1 - y) 0.3033 < y < 1The number 2.245829494 is the solution of k\ - 6k\ + 12k2 -18 = 0, 2 < Jfci<3.Example 4.3 for x, y € /,3Cp = xy + 2/? ? 6i(yK(l - x)*-\ p€[0,1]. where(X, Y) with associated copulas Cp are radially symmetric. pc0 — 3/4/?, 0 < /? < 1. This example demonstrate that one can construct copulas with 4-degree polynomial section with desired dependence p = 3/4 by Nelsen.