Copula–Induced Measures of Concordance

Abstract We study measures of concordance for multivariate copulas and copulas that induce measures of concordance. To this end, for a copula A, we consider the maps C → R given by where C denotes the collection of all d–dimensional copulas, M is the Fréchet–Hoeffding upper bound, Π is the product copula, [. , .] : C × C → R is the biconvex form given by [C, D] := ∫ [0,1]d C(u) dQD(u) with the probability measure QD associated with the copula D, and ψΛ C → C is a transformation of copulas. We present conditions on ψΛ and on A under which these maps are measures of concordance. The resulting class of measures of concordance is rich and includes the well–known examples Spearman’s rho and Gini’s gamma.


Introduction
In the present paper we study measures of concordance for copulas (of xed but arbitrary dimension) and copulas that induce measures of concordance.
The literature on measures of concordance for multivariate copulas is rich. It provides a huge number of multivariate generalizations of well-known bivariate measures of concordance (see e.g. [2, 11-13, 16, 19]), and various axiomatic de nitions of a measure of concordance (see e.g. [3,8,16]).
We here employ the quite general de nition proposed in [8]. Besides the usual normalization with respect to the Fréchet-Hoe ding upper bound M, this de nition includes two axioms which are formulated in terms of certain subgroups of the group Γ of transformations on copulas discussed in [10] for the bivariate case and in [8] for the general case. These axioms provide an easy access to the investigation of invariance properties of a measure of concordance. In particular, it turns out that for copulas which are invariant under a speci c subgroup of Γ the value of every measure of concordance is equal to zero. Since the product copula Π satis es this invariance property, we obtain the usual normalization with respect to Π.
We study measures of concordance which are induced by a xed copula. Throughout this paper, let C denote the collection of all d-dimensional copulas and consider the biconvex form [. , .] : C × C → R studied in [9], which is given by where Q D is the probability measure associated with the copula D. Consider now a copula A ∈ C and a subgroup Λ of Γ and let ψ Λ : C → C denote the map which turns every copula C to the arithmetic mean of the In what follows we present conditions on A under which these maps are measures of concordance. Thereby, we improve results given in [3,16] for the general case, and for the bivariate case we extend results from [1,5,6,10]. In addition, for a subclass of such copula-induced measures of concordance, we present lower bounds and show that these lower bounds tend to when the dimension of the copulas tends to in nity.
This paper is organized as follows: We rst recall essential de nitions and results concerning the group Γ of transformations on C discussed in [8] and the biconvex form [. , .] for copulas introduced in [9] (Section 2). We then study measures of concordance for copulas (Section 3) and copulas that induce measures of concordance (Section 4), and we present some results on lower bounds for such measures of concordance (Section 5). Auxiliary results needed for the proofs in Sections 3 and 4 are presented in the appendix (Section A).

Preliminaries
Let I := [ , ] and let d ≥ be an integer which will be kept x throughout this paper. For the sake of a concise de nition of a copula we consider, for L ⊆ { , ..., d}, the map η L : I d × I d → I d given coordinatewise by (iii) The identity C(η {i} ( , u)) = u i holds for all u ∈ I d and all i ∈ { , ..., d}. Note that, for u ≤ v, the family {η L (u, v)} L⊆{ ,...,d} consists of all vertices of the interval [u, v]. Thus, this de nition of a copula is appropriate and in accordance with the literature; see [4,14]. The collection C of all copulas is convex.

The Group Γ
A map φ : C → C is said to be a transformation. We denote by Φ the collection of all transformations and de ne the composition We now introduce two elementary transformations: For i, j, k ∈ { , ..., d} with i ≠ j we de ne the maps π i,j , ν k : C → C by letting (π i,j (C))(u) := C(η {i,j} (u, u j e i + u i e j )) where e i denotes the i-th unit vector. π i,j is called a transposition, and ν k is called a partial re ection. Both, π i,j and ν k , are involutions. There exists a smallest subgroup (Γ, •) of Φ containing all transpositions and all partial re ections. This group Γ is a representation of the hyperoctahedral group with d! d elements.
A transformation is called a permutation if it can be expressed as a nite composition of transpositions, and it is called a re ection if it can be expressed as a nite composition of partial re ections. We denote by Γ π the set of all permutations and by Γ ν the set of all re ections. Then Γ π and Γ ν are subgroups of Γ, Γ ν is commutative while Γ π is not, and every transformation in Γ can be expressed as a composition of a permutation and a re ection. Due to its particular interest we emphasize the re ection τ := ○ d k= ν k , an involution called total re ection. The total re ection τ transforms every copula into its survival copula. We set Γ τ := {ι, τ} and Γ π,τ := γ ∈ Γ γ = π • φ for some π ∈ Γ π and some φ ∈ Γ τ . Then Γ τ is the center of Γ, and Γ π,τ is a subgroup of Γ.
For a subgroup Λ of Γ, a copula C is called Λ-invariant if it satis es γ(C) = C for every γ ∈ Λ. The copula M is Γ π,τ -invariant and the copula Π is Γ-invariant. For the ease of notation, we put

The Biconvex Form
For a convenient study of measures of concordance, we use the map [. , .] : C × C → R given by where Q D denotes the probability measure associated with the copula D. The map [. , .] is linear with respect to convex combinations in both arguments and is therefore called a biconvex form. Moreover, [. , .] is monotonically increasing in both arguments with regard to the concordance order c on C (i.e. C ≤ D and τ(C) ≤ τ(D)), and it satis es ≤ [C, D] ≤ [M, M] = / for all C, D ∈ C (note that C c M for all C ∈ C); we also note that [Π, Π] = / d .

Measures of Concordance
In this section, we study measures of concordance for copulas. We employ the quite general de nition of a measure of concordance proposed in [8], in which the axioms are formulated in terms of two particular subgroups of the group Γ, and we discuss invariance properties following from these axioms.
A map κ : C → R is said to be a measure of concordance if it satis es the following axioms: (ii) The identity κ(γ(C)) = κ(C) holds for all γ ∈ Γ π,τ and all C ∈ C. For the case d = , this de nition is in accordance with [5-7, 10, 15].
Axiom (ii) implies that every transformation in Γ π,τ preserves the value of every measure of concordance. Moreover, it turns out that the transformations in Γ π,τ are in fact the only transformations in Γ having this property; see [8, Theorem. 6.2.]: Proposition 3.1. Let κ be a measure of concordance and γ ∈ Γ. Then the identity κ(γ(C)) = κ(C) holds for every C ∈ C if, and only if, γ ∈ Γ π,τ . Axiom (iii) provides a simple test on copulas under which every measure of concordance of a copula is equal to zero: Corollary 3.2. Let κ be a measure of concordance. Then the identity κ(C) = holds for every Γ ν -invariant copula C ∈ C. In particular, κ(Π) = .
Thus, every measure of concordance is normalized by the copulas M and Π.

Copula-Induced Measures of Concordance
In the present section, we discuss a class of measures of concordance which are de ned in terms of the biconvex form [. , .], a subgroup Λ of Γ and a xed copula A ∈ C. More precisely, we consider the map κ for all C ∈ C. We also note that, if Λ ⊆ Γ π,τ , both maps satisfy axiom (i) of a measure of concordance due to the fact that M is Γ π,τ -invariant. Without any assumptions on A we obtain the following result whose proof is straightforward: In what follows we present conditions on A under which κ • A,Λ and κ • A,Λ are measures of concordance. The following proposition summarizes the results for the bivariate case which are essentially due to [5,6]; see also [10]. Note that C {ι} = C for all C ∈ C.
which contradicts axiom (ii) of a measure of concordance. Indeed, if d is odd, then the copula C with To obtain measures of concordance in the general case, one may replace the trivial subgroup {ι} by an appropriate subgroup Λ of Γ with Λ ⊆ Γ π,τ and impose a condition on A. For example, one has the following result which is essentially due to [3,16]:

Proposition 4.4. Assume that A is Γ-invariant. Then each of the maps κ • A,Γ τ and κ •
A,Γ τ is a measure of concordance.
The following examples are multivariate generalizations of Gini's gamma and Spearman's rho:  Proposition 4.4 implies that Gini's gamma is a measure of concordance. The de nition of Gini's gamma used here is in accordance with that in [16].  Proposition 4.4 implies that Spearman's rho is a measure of concordance. The de nition of Spearman's rho used here is in accordance with that in [13].
Since Γ-invariance of a copula is a quite strong property, one would like to relax the assumption on A. To this end, we replace the arithmetic mean under Γ τ by the arithmetic mean under the larger subgroup Γ π,τ used in axiom (ii) and Proposition 3.
for all C ∈ C. Moreover, we recall that κ • A,Γ π,τ and κ • A,Γ π,τ satisfy axiom (i) of a measure of concordance, and it follows from Lemma A.1 that they satisfy axiom (ii) as well. We are hence interested in conditions on A under which each of these maps also satis es axiom (iii) of a measure of concordance. We start our investigation with a quite abstract characterization:
for all C ∈ C.
We now prove (2). To this end, assume that the identity [C Γ ν , and the above identity then implies for all C ∈ C. Thus, κ • A,Γ π,τ satis es axiom (iii) of a measure of concordance. Assume now that [M, A] ≠ [Π, Π] and that κ • A,Γ π,τ is a measure of concordance. Then the above identity implies for all C ∈ C. Thus, the identity [C Γ ν , A Γ π ] = [Π, Π] holds for all C ∈ C. This proves (2). Finally, let κ • A,Γ π,τ be a measure of concordance. Then (2) and Lemma A.
At this point, we brie y discuss the condition on A used in Theorem 4.7; the proof of Lemma 4.9 is straightforward: The converse implication of Lemma 4.9 (1) is not true in general; see Example 4.8 (A ≠ ν (A)). The following example shows that also the converse implication of Lemma 4.9 (2) is not true, in general: Example 4.10. The function A : I d → I given by We recall that the copula A Γ ν is Γ ν -invariant, and it follows from Lemma 4.9 that the copula (A Γ ν ) Γ π is Γ νinvariant. Therefore, Theorem 4.7 implies that, via a suitable transformation, even the copula A induces a measure of concordance.
For the case where A is Γ-invariant, we obtain the following improvement of the assertions of Proposition 4.4 and Theorem 4.7: Proof. Lemma A.2 and Lemma A.
We conclude our study considering the case d = . Proposition 4.2 implies the following result which relaxes the condition on A to induce a measure of concordance: We now prove (2). To this end, assume that A Λ is Γ-invariant. Then Lemma A.4 (1) and Proposition 4.