A biconvex form for copulas

Abstract We study the integration of a copula with respect to the probability measure generated by another copula. To this end, we consider the map [. , .] : C × C → R given by where C denotes the collection of all d–dimensional copulas and QD denotes the probability measures associated with the copula D. Specifically, this is of interest since several measures of concordance such as Kendall’s tau, Spearman’s rho and Gini’s gamma can be expressed in terms of the map [. , .]. Quite generally, the map [. , .] can be applied to construct and investigate measures of concordance.


Introduction
In the present paper we study the map [. , .] : C × C → R given by where C denotes the collection of all d-dimensional copulas and Q D denotes the probability measures associated with the copula D. A probability measure generated by a copula is said to be a copula measure. The map [. , .] is linear with respect to convex combinations in both arguments and is therefore called a biconvex form.
For dimension d = , integration of a copula with respect to a copula measure has been frequently studied in connection with measures of concordance (see e.g. [8][9][10][11][12]14]), with respect to transformations of copulas (see e.g. [3]), and also with regard to applications to the joint distribution of random vectors (see [7]). For d = the biconvex form [. , .] is symmetric, but it turns out that this property gets lost in higher dimensions.
Here we discuss the biconvex form [. , .] for an arbitrary but xed dimension d ≥ . We show that the map [. , .] is bounded by from below and by / from above, and that it is monotone with regard to concordance order. We further study the biconvex form with respect to the group Γ of transformations on C discussed in [6] for the bivariate case and in [5] for the general case. The group Γ contains an involution τ which transforms every copula into its survival copula, and it turns out that this transformation changes the arguments of the biconvex form. Moreover, we identify those transformations γ ∈ Γ which satisfy [γ(C), γ(D)] = [C, D] for all C, D ∈ C, and we consider a class of copulas D such that [C, D] = / d holds for all C, D ∈ D.
Several measures of concordance such as Kendall's tau κτ and Spearman's rho κρ can be expressed in terms of the biconvex form [. , .]. Namely, using the Fréchet-Hoe ding upper bound M and the product copula Π *Corresponding Author: Sebastian Fuchs: Lehrstuhl für Versicherungsmathematik, Technische Universität Dresden, E-mail: sebastian.fuchs1@tu-dresden.de (see [11]). However, the biconvex form [. , .] and its properties with regard to the transformations in Γ can not only be employed to construct but also to investigate measures of concordance (see e.g. [1,15]).
This paper is organized as follows: We rst recapitulate essential de nitions and results concerning copulas, copula measures and the integration with respect to a copula measure (Section 2). In Section 3 we then introduce and study the biconvex form [. , .] with regard to bounds, symmetry, convergence and order. It turns out that the transformation τ provides a useful tool which helps us to prove that certain properties which are valid for one of the arguments of the map [. , .] are also valid for the other argument (Section 4). Finally, we investigate the biconvex form [. , .] with regard to transformations in Γ (Section 5).

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 A copula is a function C : I d → I satisfying the following conditions: (ii) The identity C(η i (u, )) = holds for all u ∈ I d and all i ∈ { , ..., d}. (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 [2,12]. 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 ∈ { , ..., d} with i ≠ j we de ne the map π i,j : C → C by letting (π i,j (C))(u) := C(η {i,j} (u, u j e i + u i e j )) and, for k ∈ { , ..., d}, the map ν k : C → C by letting (ν k (C))(u) := C(η k (u, )) − C(η k (u, − u)) 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. Then there exists a smallest subgroup (Γ, •) of Φ containing all transpositions and all partial re ections. A transformation is called a permutation if it can be expressed as a nite composition of transpositions, and a transformation 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 whereas only Γ ν is commutative, 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. 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. The total re ection τ transforms every copula into its survival copula.
A copula C is called invariant with respect to a subgroup Λ of Γ (or Λ-invariant) if it satis es γ(C) = C for every γ ∈ Λ.
The group Γ is a representation of the hyperoctahedral group with d! d elements. Note that the hyperoctahedral group has another representation which is geometric and quite popular; see [15][16][17]: Consider the collection of all vector-valued functions from I d into I d equipped with the composition and the identityι. Then there is a smallest group (Γ, ) containing the vector-valued functionsπ i,j : Note that everyγ ∈Γ can be expressed as a nite composition ofπ i,j andν k with i, j, k ∈ { , ..., d} and i ≠ j. Sinceπ i,j andν k are continuous we hence obtain that everyγ ∈Γ is continuous as well. For the composition of n ∈ N functionsγm ∈Γ, m ∈ { , ..., n}, we write n ♦ m= γm := ι n = γn ♦ n− m= γm otherwise and, for N = { , ..., n} and a set of pairwise commutingγm ∈Γ, m ∈ N, we put ♦ m∈Nγm := ♦ n m= γm. We haveτ : The groups Γ andΓ are related to each other by an isomorphism T : For a detailed discussion of the groups (Γ, •) and (Γ, ), see [5]. We denote by Q the collection of all copula measures. Then Q is convex. Since the functions inΓ are continuous and hence measurable, they can be used to transform copula measures, and it turns out that Q is stable under the transformations of the groupΓ; see [5,Theorem 7.3.]. The next result is a re nement of the correspondence theorem relating distribution functions on R d to probability measures on B(R d ); see [5, Theorem 7.2]:

Theorem.
There exists a one-to-one correspondence S : C → Q. Moreover, every corresponding pair of a copula C ∈ C and a copula measure Q ∈ Q satis es C Moreover, we have the following representation including the isomorphism T and the one-to-one correspondence S; Theorem 2.2 follows from the measure extension theorem: The previous result can be represented by the following commuting diagram: For the ease of notation we write Q C := S(C). As a consequence of Theorem 2.2 we can formulate the substitution rule for copula measures:

Lemma. For every C ∈ C and every γ ∈ Γ the identity
holds for all positive measurable functions f : I d → R.
We nally discuss some useful results concerning the integration of a positive measurable function with respect to a copula measure. Lemma 2.4 is due to Fubini's Theorem and the fact that every copula measure has uniform margins.

Lemma. For every C ∈ C the identity
holds for all i ∈ { , ..., d}.

Examples.
(1) The copula measure of M satis es holds for all positive measurable functions f : I d → R.

A Biconvex Form for Copulas
In the present section we study the integration of a copula with respect to a copula measure. To this end, we introduce a biconvex form on C × C which we investigate with regard to bounds, symmetry, convergence and order.
Note that a copula is a positive measurable function. We de ne the map [. , .] : C × C → R by letting The map [. , .] is linear with respect to convex combinations in both arguments and is therefore called a biconvex form. For the copulas M and Π we have the following results: The next examples assert that, for any d ≥ , the map [. , .] fails to be symmetric:

Examples.
(1) Consider d ∈ N + . Then the function E : I d → R given by is a copula (see [1,Example 4.5]) and satis es (2) Consider d ∈ N + . Then, for j ∈ { , ..., d}, the function E : I d → R given by is a copula (see [5, Proof of Theorem 3.1]) and satis es The following result deals with convergence of the biconvex form [. , .]; it immediately follows from the dominated convergence theorem: The next examples assert that, for any d ≥ , the map D → [Π, D] fails to be monotonically increasing with regard to the pointwise order relation:

Examples.
(1) Consider d ∈ N + . Then the copula E discussed in Example 3.4 (1) satis es Π ≤ E, but (2) Consider d ∈ N + . Then the copula E discussed in Example 3.4 (2) satis es Π ≤ E, but We conclude this section with the following result, whose proof for d = is due to [4]. Then Cv,ε satis es Cv,ε(u) = Π(u) + ε d for all u ∈ Mε and Cv,ε(u) = Π(u) for all u ∈ I d \( , v). We prove that Cv,ε is a copula: For every u, v ∈ I d such that u ≤ v we obtain K⊆{ ,...,d} for all C, D ∈ C. This proves the assertion.

The Biconvex Form and Γ
In the present section we investigate the biconvex form [. , .] with respect to the transformations in Γ. We rst identify those transformations γ ∈ Γ which satisfy [γ(C), γ(D)] = [C, D] for all C, D ∈ C. Finally, we consider a class of copulas D such that [C, D] = / d holds for all C, D ∈ D.
Proof. The result is immediate from Lemma 2.3.