© Hindawi Publishing Corp. MEASURES OF CONCORDANCE DETERMINED BY D4-INVARIANT COPULAS

A continuous random vector ( X , Y ) uniquely determines a 
copula C : [ 0 , 1 ] 2 → [ 0 , 1 ] such that when the distribution 
functions of X and Y are properly composed into C , the 
joint distribution function of ( X , Y ) results. A copula is 
said to be D 4 -invariant if its mass distribution is 
invariant with respect to the symmetries of the unit square. 
A D 4 -invariant copula leads naturally to a family of 
measures of concordance having a particular form, and all 
copulas generating this family are D 4 -invariant. The 
construction examined here includes Spearman’s rho and 
Gini’s measure of association as special cases.


Introduction. Let I = [0, 1] and
. µ is a doubly stochastic measure on I 2 if it is a probability measure on the Borel sets of I 2 such that µ(A×I) = µ(I ×A) = λ(A), where A is a Borel set of I and λ is the one-dimensional Lebesgue measure.A copula (more precisely a 2-copula) is a function C : I 2 → I that is related to some doubly stochastic measure, µ, by C(x, y) = µ([0,x] × [0,y]) (see [3]).There is a one-to-one correspondence between copulas and doubly stochastic measures.
Besides being associated with a doubly stochastic measure, a copula can be uniquely determined by a continuous random vector.By Sklar's theorem, for any continuous random vector, (X, Y ), with marginals, F X and F Y , respectively, and joint distribution function, F X,Y , there exists a unique copula, C, such that F X,Y (x, y) = C(F X (x), F Y (y)) (see [1,3]).
The simplest examples are as follows.If Y is an almost surely increasing function of X, then its associated copula is M(x, y) = min(x, y).If Y is an almost surely decreasing function of X, then its associated copula is W (x,y) = max(x + y − 1, 0).Finally, if X and Y are independent, then their associated copula is Π(x, y) = xy (again see [3]).
When considering two random variables, it can be useful to know how much large values of one random variable correspond to large values of the other.More formally, for any two observations, (X 1 ,Y 1 ) and (X 2 ,Y 2 ), from a continuous random vector, (X, Y ), the two observations are said to be concordant if either X 1 < X 2 and Y 1 < Y 2 , or X 2 < X 1 and Y 2 < Y 1 .Similarly, the two observations are said to be discordant if either The properties of concordance and discordance can be gauged by a measure of concordance, a concept developed by Scarsini [4] and presented in [3].
A measure of concordance associates to a continuous random vector, (X, Y ), a real number, κ X,Y .As developed by Scarsini, it can be shown that this value depends only on the copula C, uniquely associated with (X, Y ).Because of this, we sometimes write κ C instead of κ X,Y .The following definition of a measure of concordance can be found in [3].Definition 1.1.Let C be the copula associated with the continuous random vector, (X, Y ).Let κ X,Y be a functional mapping the set of all copulas to R. κ X,Y (which can also be denoted κ C if C is the copula for (X, Y )) is a measure of concordance if the following conditions are satisfied: (1) κ C is defined for every copula, C, Spearman's rho, ρ, and Gini's measure of association, γ, are two examples of measures of concordance.Spearman's rho can be expressed as ρ C = 12 I 2 CdΠ − 3, where Π(x, y) = xy, while Gini's measure of association can be expressed as γ C = 8 I 2 Cd((M + W )/2)−2, where M(x, y) = min(x, y) and W (x,y) = max(x +y −1, 0) [3,5].Note that each is of the form κ C = α I 2 CdA − β, where A is a fixed copula and α, β ∈ R. Definition 1.2.A copular measure of concordance is one of the form κ C = α I 2 CdA − β, where A is a fixed copula and α, β ∈ R.
When you are dealing with copular measures of concordance you are in effect dealing with an expression where the difference of the probabilities of concordance and discordance are taken.Namely, for any continuous random vectors, (X 1 ,X 2 ) and (Y 1 ,Y 2 ), respectively, associated with a copula C and a fixed, copular-generating copula A, we are dealing with P (( For more details on this matter, one may refer to [3].
The standard notation for the group of symmetries on the unit square I 2 is D 4 .We have D 4 = {e, r , r 2 ,r 3 ,h,hr ,hr 2 ,hr 3 }, where e is the identity, h is the reflection about x = 1/2, and r is a 90 • counterclockwise rotation.
For d ∈ D 4 , a new copula, C d , can be formed, where While it might not always be obvious that a copula the principles behind the main theorem take shape, giving a nice theoretical characterization and providing a way to construct many measures of concordance.The theorem states that a copula is copular generating if and only if it is D 4 -invariant.
In the second section, some background information is given, where measures of concordance are considered entirely in terms of copulas and their symmetries.Also included in the section are some helpful lemmas with their proofs.The third and final section includes the formulation and proof of the main result, in addition to some remarks we think may be of some interest.

Background and lemmas.
Here and in all that follows, we assume that we are dealing with continuous random vectors.
Observe Table 2.1 with regard to the correspondence between the copula C d for each d ∈ D 4 and a random vector with which it is associated.Note that When considering the copulas M, W , and Π as well as Table 2.1, Definition 1.1 may be rewritten.Definition 2.1.Let C be the copula associated with the continuous random vector, (X, Y ).Let κ C be a functional mapping the set of all copulas to R. κ C is a measure of concordance if the following conditions are satisfied: (1) κ C is defined for every copula C, Proof.Let (X 1 ,Y 1 ) and (X 2 ,Y 2 ) be independent, continuous random vectors associated with A and B, respectively.Since BdA. (2.1) ,hr ,hr 3 } and hG = {h, hr 2 ,r ,r 3 }.Given copulas A and B, the following are true: (1) ) and (X 2 ,Y 2 ) be independent, continuous random vectors associated with A and B, respectively.
For d = h, (2.2) For d = r 2 using Lemma 2.2, (2.3) Noting that hr = r 3 h and r 2 is in the center of D 4 , we then have for d = r , AdB r . (2.4) Since our assertion holds for d = r ,r 2 , the case when d = r 3 is clear.Since r 2 is in the center of D 4 and our assertion holds for d = h, r 2 , the case when d = hr 2 is readily seen.
For d = hr , Finally, for d = hr 3 , (2.6) Consider a grid being placed on I 2 such that it is divided into square cells having the dimensions 1/n × 1/n.We construct a copula by assigning a constant mass density, δ i,k , to the cell in the ith column from the left and kth row from the bottom, where Such a notion is a particular instance of a checkerboard copula (see [1]).
The following concepts and notation will be used to construct the checkerboard copulas, Q 1 p,n and Q 2 p,n , that depend on a fixed point p ∈ (0, 1) 2 and n ∈ N.
(ii) Choose p in the interior of I 2 .There exists N ∈ N such that for p = (p 1 ,p 2 ), 1/N < min(p 1 ,p 2 ) and Np 1 ,Np 2 ∉ N. Let ᏺ be an infinite, increasing sequence of such N.
(iii) Let Q 1 p,n and Q 2 p,n be two n × n checkerboard copulas, where n ∈ ᏺ, and having density δ 1 i,k and δ 2 i,k , respectively, in cell J i,k .(iv) The cell containing p will be denoted J i * ,k * .(v) Let the Q 1 p,n have the following densities assigned to its cells: (2.7) (vi) Let Q 2 p,n have the following densities assigned to its cells: (2.8) We make use of Q 1 p,n and Q 2 p,n in some of the following proofs. (2.9) By the mean value theorem, there exists p a,b ∈ J a,b such that (2.10) ). Letting n → ∞, since either the x coordinate, y coordinate, or both coordinates of p 1,1 , p i * ,1 , and p 1,j * will go to 0, it follows from the continuity of A and B that (2.12) Proof.Note that since A, A h , and A hr are all copulas, A(p) = A h (p) = A hr (p) for every p on the boundary of I 2 .Because of this, only p in the interior of I 2 needs to be considered.Using By subtraction and application of Lemmas 2.2 and 2.3, we have (2.14) so that (2.15) Finally, by using the same argument as in Lemma 2.4, the results A = A h and A = A hr are attained.Since h and hr generate D 4 , we know A is D 4 -invariant.

Theorem 3.1. A copula is copular generating if and only if it is D 4 -invariant.
Proof.Suppose that A is copular generating.Note that α ≠ 0 since a measure of concordance is not constant.Therefore, by Lemma 2.5, A is D 4 -invariant.Now, we assume that A is D 4 -invariant.Setting where α = ( I 2 MdA − 1/4) −1 , we will show that κ is a measure of concordance.It needs to be shown that I 2 MdA − 1/4 ≠ 0 in order for κ C to be defined for every copula C. Noting by the By the chosen form of κ, it is clear that κ M = 1.By the D 4 -invariance of A and Lemma 2.3, It is similarly attained that κ C h = κ C hr 2 = −κ C .In particular, noting that M h = W and Π h = Π, we see that κ W = −1 and κ Π = 0. Recall from (3.2) that α > 0. Since Finally, since every sequence of copulas converging to a copula pointwise does so uniformly (see [3]), it follows that if C n → C pointwise, then Remark 3.2.By Theorem 3.1, any D 4 -invariant copula and only a D 4 -invariant copula generates a copular measure of concordance.For example, one may generate a copular measure of concordance using the copula associated with the circular uniform distribution which is presented in [3]: , otherwise. (3.4)

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: gives the amount of probabilistic mass contained in the rectangle d([0,x] × [0,y]) as determined by the doubly stochastic measure associated with C. Definition 1.4.A copula is D 4 -invariant if for every d ∈ D 4 , C(x, y) = C d (x, y) for all (x, y) ∈ I 2 .

Lemma 2 . 4 .
If for copulas A and B, I 2 AdC = I 2 BdC for every copula C, then A = B. Proof.For convenience, we write I 2 (A − B)dC = 0 for every copula C. Since A and B are copulas, A(p) = B(p) for any p on the boundary of I 2 .Thus, only p in the interior of I 2 needs to be considered.Using Q 1 p,n and Q 2 p,n as choices for C yields 0 .9) Thus, by Lemma 2.2,I 2 ( α Â − αA)d(Q 1 p,n − Q 2 p,n ) = 0.Using the same argument as in Lemma 2.4 results in α Â(p) = αA(p) for any p ∈ (0, 1) 2 .For p = (p 1 ,p 2 ), letting p 1 → 1 or p 2 → 1, the uniform margins and continuity of Â and A force α = α and consequently, β = β.Thus, I 2 ÂdC = I 2 AdC for every copula C, which shows that Â = A by Lemma 2.4.

Table 2 .
1. Symmetries of copulas on I 2 and their associated random vectors.
For any copulas A and B, I 2 AdB = I 2 BdA.