On Minimal Copulas under the Concordance Order

In the present paper, we study extreme negative dependence focussing on the concordance order for copulas. With the absence of a least element for dimensions $d\ge$ 3, the set of all minimal elements in the collection of all copulas turns out to be a natural and quite important extreme negative dependence concept. We investigate several sufficient conditions and we provide a necessary condition for a copula to be minimal: The sufficient conditions are related to the extreme negative dependence concept of d-countermonotonicity and the necessary condition is related to the collection of all copulas minimizing multivariate Kendall's tau. The concept of minimal copulas has already been proved to be useful in various continuous and concordance order preserving optimization problems including variance minimization and the detection of lower bounds for certain measures of concordance. We substantiate this key role of minimal copulas by showing that every continuous and concordance order preserving functional on copulas is minimized by some minimal copula and, in the case the continuous functional is even strictly concordance order preserving, it is minimized by minimal copulas only. Applying the above results, we may conclude that every minimizer of Spearman's rho is also a minimizer of Kendall's tau.


Introduction
The strongest notion of positive dependence is given by comonotonicity. A d−dimensional continuous random vector is said to be comonotonic if one of the following equivalent conditions is satisfied; see, e.g., [9]: -the random vector has the upper Fréchet-Hoeffding bound M as its copula, -each of its coordinates is almost surely an increasing transformation of the other, -each of its bivariate subvectors is comonotonic. In many applications in finance and insurance, comonotonicity can be regarded as one of the most dangerous behaviours and, with its popularity in risk management as an extreme positive dependence, it seems quite reasonable that also extreme negative dependence is getting more attention, see, e.g., [5]. In the bivariate case, the strongest notion of negative dependence is given by countermonotonicity. A bivariate continuous random vector is said to be countermonotonic if one of the following equivalent conditions is satisfied; see, e.g., [9]: -the random vector has the lower Fréchet-Hoeffding bound W as its copula, -each of its coordinates is almost surely a decreasing transformation of the other. In contrast to comonotonicity, for dimensions d ≥ 3 there exist no continuous random vector for which all the bivariate subvectors are countermonotonic and hence there exists no single agreed definition of extreme negative dependence in arbitrary dimension; see, e.g., [3; 4; 16; 21; 39; 40].
In the present paper, we investigate extreme negative dependence focussing on the concordance order for copulas. It is well-known that M is the greatest element and, in the bivariate case, W is the least element in the collection of all copulas with respect to concordance order. With the absence of a least element for dimensions d ≥ 3, the set of all minimal elements (so-called minimal copulas), i.e. all locally least elements, in the collection of all copulas turns out to be a natural and quite important extreme negative dependence concept, in particular, with regard to the minimization of continuous and concordance order preserving optimization problems. This includes several measures of dependence like Kendall's tau and Spearman's rho, but also the variance of the sum of several given random variables as a map on copulas; see, e.g., [6]. While it is well-known that every continuous and (strictly) concordance order preserving functional on copulas is (uniquely) maximized by M and, in the bivariate case, (uniquely) minimized by W , less is known about minimization for dimensions d ≥ 3. Most recently, Ahn [1] and Lee et al. [22] have demonstrated the potential of minimal copulas in variance minimization when the marginals are uniform, elliptical or belong to the unimodal-symmetric location-scale family; we refer to [31; 39] for further results on variance minimization. Moreover, Genest et al. [18] and Lee and Ahn [21] have provided minimal copulas minimizing certain measures of concordance. Further continuous and concordance order preserving functionals on copulas are discussed in [6; 14], but also in [23; 28; 29; 35] where the necessary properties of the functional strictly depend on the function to be integrated; for more details on this topic we refer to [29] and the references therein.
In this paper, we first discuss the existence and list some important examples of copulas that are minimal with respect to concordance order. We then investigate several sufficient conditions (Proposition 3.2 and Theorem 3.6) and we provide a necessary condition (Theorem 3.9) for a copula to be minimal. The sufficient conditions are related to the extreme negative dependence concept of K−countermonotonicity introduced in [21 ; 22] and the necessary condition is related to the collection of all copulas minimizing multivariate Kendall's tau: It turns out that every minimal copula minimizes Kendall's tau which is the main result of this paper. We further point out the key role of minimal copulas with regard to the minimization of continuous and (strictly) concordance order preserving optimization problems: It turns out that every continuous and concordance order preserving functional on copulas is minimized by some minimal copula and that, in the case the continuous functional is even strictly concordance order preserving, it is minimized by minimal copulas only (Theorem 4.5). Finally, we apply our results to Kendall's tau and Spearman's rho and show that every minimizer of Spearman's rho is also a minimizer of Kendall's tau (Corollary 5.5).

Preliminaries
In this section, we fix some notation and recall some definitions and results on copulas, a group of transformations of copulas and the concordance order.
Let I := [0, 1] and let d ≥ 2 be an integer which will be kept fix throughout this paper. We denote by e 1 , . . . , e d the unit vectors in R d , by 0 the vector in R d with all coordinates being equal to 0 and by 1 the vector in R d with all coordinates being equal to 1. For u, v ∈ R d , we use the notation u ≤ v resp. u < v in the usual sense such that u k ≤ v k resp. u k < v k holds for every k ∈ {1, . . . , d}.
(iii) C(η k (1, u)) = u k holds for every k ∈ {1, ..., d} and every u ∈ I d . This definition is in accordance with the literature; see, e. g., [9; 26]. The collection C of all copulas is convex. The following copulas are of particular interest: -The upper Fréchet-Hoeffding bound M given by M(u) := min{u 1 , . . . , u d } is a copula and every copula C satisfies C(u) ≤ M(u) for every u ∈ I d . -The product copula Π given by Π(u) := d k=1 u k is a copula. -The lower Fréchet-Hoeffding bound W given by W (u) := max{ d k=1 u k + 1 − d, 0} is a copula only for d = 2, and every copula C satisfies W (u) ≤ C(u) for every u ∈ I d .
Since every copula C has a unique extension to a distribution function R d → I, there exists a unique probability measure Q C : The probability measure Q C is said to be the copula measure with respect to C and it

A Group of Transformations of Copulas
Let Φ denote the collection of all transformations C → C and consider the composition • : Φ × Φ → Φ given by (ϕ 2 • ϕ 1 )(C) := ϕ 2 (ϕ 1 (C)) and the map ι ∈ Φ given by ι(C) := C. Then (Φ, •) is a semigroup with neutral element ι. For i, j, k ∈ {1, ..., d} with i = j, we define 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 )) (ν k (C))(u) := C(η k (u, 1)) − C(η k (u, 1−u)) Each of these maps is an involution and there exists -a smallest subgroup Γ π of Φ containing every π i,j , -a smallest subgroup Γ ν of Φ containing every ν k and -a smallest subgroup Γ of Φ containing Γ π ∪ Γ ν . The transformations in Γ π are called permutations and the transformations in Γ ν are called reflections. The group Γ ν is commutative and for K ⊆ {1, ..., d} we define . We note that the total reflection τ := ν {1,...,d} transforms every copula into its survival copula. From a probabilistic viewpoint, if U is a random vector whose distribution function is the copula C, then the reflected copula ν K (C) of C is the distribution function of the random vector η K (U, 1 − U). We refer to [13] for further details on the groups Γ π , Γ ν and Γ.

Concordance Order
A relation ⋖ on C is said to be an order relation if it is reflexive, antisymmetric and transitive; in this case the pair (C, ⋖) is called ordered set. The copula C ∈ C is said to be -the greatest element, if the inequality D ⋖ C holds for all D ∈ C.
-the least element, if the inequality C ⋖ D holds for all D ∈ C.
-a maximal element, if, for every D ∈ C, the inequality C ⋖ D implies C = D.
-a minimal element, if, for every D ∈ C, the inequality D ⋖ C implies C = D. We denote by m(C, ⋖) the set of all minimal elements of (C, ⋖).
Then is an order relation which is called the concordance order on C. Since τ (M) = M, the upper Fréchet-Hoeffding bound M is the greatest element in (C, ); similarly, in the case d = 2, the lower Fréchet-Hoeffding bound W is the least element in (C, ). A copula C ∈ C is said to be a minimal copula if C ∈ m(C, ).

Multivariate Countermonotonicity
With the absence of a least element for dimensions d ≥ 3, the set of all minimal elements, i.e. all locally least elements, in the collection of all copulas turns out to play an important role when studying extreme negative dependence concepts. In this section, we discuss the existence and list some important examples of copulas that are minimal with respect to concordance order (so-called minimal copulas). Additionally, we investigate several sufficient conditions and we provide a necessary condition for a copula to be minimal.
First sufficient and necessary conditions for a copula to be minimal can be achieved by comparing concordance order with other order relations: 3.1 Remark. Let ⋖ be an order relation on C.
Note that (1) is applicable to pointwise order and (2) is applicable to supermodular order; for more details on the comparison of concordance order with other order relations, we refer to [20; 24].
We are now interested in sufficient and necessary conditions for a copula to be minimal that are formulated in terms of the copula itself.
-a family {g k } k∈K of strictly increasing and continuous functions I → R and -some c ∈ R such that Proof. The inclusions were proved in [22,Theorem 4,Lemma 5], and the equivalence for d = 2 follows from Example 3.4 (1) below.

Examples 3.4 below show that (1) the inclusion in Proposition 3.2 for
the inclusion in Proposition 3.2 for |K| = d − 1 is strict, and that (3) the inclusion in Proposition 3.2 fails to be satisfied whenever 2 ≤ |K| ≤ d − 2.
In the following, we list some important minimal copulas and show in passing that the set m(C, ) is non-empty: is d−CM and hence minimal.
for every u ∈ I d satisfying η K (u, 1 − u) = α1 for some α ∈ I, and hence The assertion then follows from Proposition 3.2. (2) The copula C given by is Archimedean and is called the Clayton copula with parameter −1/(d − 1). It follows from [21,Theorem 4] and Proposition 3.2 that C is d−CM and hence minimal.
Then, by [34, Theorem 6.6.3], D is a d-dimensional copula and it follows from Proposition 3.2 that D is minimal. However, D fails to be d−CM which follows from straightforward calculation. (2) Consider d ≥ 3. Then the copula C discussed in Example 3.3 (2) is d−CM and hence minimal. However, it is evident that C fails to be K−CM whenever 2 ≤ |K| ≤ d − 1.
We proceed with the discussion of a necessary condition for a copula to be minimal.
holds for every u ∈ (0, 1). We denote by C τ −CM the collection of all copulas that are τ -CM. The term Kendall-countermonotonicity is motivated by the fact that a copula C is τ -CM if and only if C minimizes multivariate Kendall's tau; see [16]. Kendall's tau is a map κ : C → R given by and the definition of Kendall's tau is in accordance with [25]. The following characterization of Kendall-countermonotonicity is due to [16, Lemma 3.1, Theorem 3.4]: 3.7 Proposition. For a copula C ∈ C the following are equivalent: It follows from [16, Theorem 3.3] that, for d = 2, a copula C is τ -CM if and only if C = W . Thus, Kendall-countermonotonicity may be regarded as a natural extension of countermonotonicity to dimensions d ≥ 3.

Remark. A subset
Consider now a copula C for which there exists some strictly comonotonic set A ⊆ (0, 1) consisting of at least two points such that i.e. the support supp Q C of Q C contains some strictly comonotonic subset of (0, 1). It then follows from Proposition 3.7 (3) that such a copula C fails to be Kendall-countermonotonic.
Thus, one may interpret Kendall-countermonotonicity as the one extreme negative dependence concept where it is inadmissible for a copula to have some strictly comonotonic support.
The following theorem is the main result of this paper; it states that every minimal copula is τ -CM and hence every minimal copula minimizes Kendall's tau.
3.9 Theorem. We have Proof. The inclusion is proved in the appendix (see Lemmas A.1, A.2 and A.3); there, for a copula C ∈ C\C τ −CM , we construct a copula D ∈ C satisfying D C with D = C which then implies that C is not minimal. The identity for d = 2 follows from [16, Theorem 3.3].
The following example shows that the inclusion in Theorem 3.9 is strict whenever d ≥ 4: It is interesting to note that also every K−CM copula is τ −CM: 3.11 Proposition. The inclusion Proof. The inclusions follow from [21,Theorem 6], and the equivalence for d = 2 is a consequence of Example 3.12 below.
The following example shows that the inclusions in Proposition 3.11 are strict whenever d ≥ 3: 3.12 Example. For d ≥ 3, consider the copula Then, C is τ -CM but fails to be K-CM whenever 2 ≤ |K| ≤ d.
such that, due to Example 3.12, at least one of the inclusions is strict.

Continuous and Order Preserving Functionals
In this section we study minimal copulas in connection with continuous and concordance order preserving functionals and show that each such optimization problem is minimized by some minimal copula. In particular, we show that any continuous functional that is even strictly concordance order preserving is minimized by minimal copulas only.
A map κ : C → R is said to be continuous if, for any sequence {C n } n∈N ⊆ C and any copula C ∈ C, uniform convergence lim n→∞ C n = C implies lim n→∞ κ(C n ) = κ(C). concordance order preserving if the inequality κ(C) ≤ κ(D) holds for all C, D ∈ C satisfying C D. strictly concordance order preserving if it is concordance order preserving and the strict inequality κ(C) < κ(D) holds for all C, D ∈ C satisfying C D with C = D.
We start with the discussion of some topological properties of C: It is well-known that C is a compact subset of the space (Ξ(I d ), d ∞ ) of all continuous real-valued functions with domain I d under the topology of uniform convergence; see, e.g., [9, Theorem 1.7.7]. It is further well-known that the range of C with respect to any continuous map κ : C → R is compact in R; see, e.g., [30,Theorem 4.14]. Even though a continuous functional C → R is minimized by some copula C ∈ C, the calculation of its minimal value can be quite difficult. For an illustration, let us consider the following quite popular optimization problem for which a solution was recently presented in [39]: 4.2 Example. The map κ : C → R given by is a continuous and strictly concordance order preserving functional, and thus, due to Proposition 4.1, κ is minimized by some copula C ∈ C. In [39, Corollary 4.1 and Figure 3.2] the authors have presented a solution for inf C∈C κ(C) in arbitrary dimension and, for d = 3, a minimal copula minimizing κ.
We now show that, for any compact subset D ⊆ C, the ordered set (D, ) is coverable from below, i.e. for every copula D ∈ D, there exists some minimal copula C ∈ (D, ) satisfying C D.

Theorem.
If D ⊆ C is compact, then (D, ) is coverable from below. In particular, (C, ) is coverable from below.
Proof. Consider D ∈ D and define the subset E ⊆ D by letting E := E ∈ D E D .
Since D ∈ E, the set E is non-empty. We first show that E is a compact subset of (D, d ∞ ).
Since E is a subset of the compact set D, it is enough to show that E is closed; compare To show that E 0 is a minimal copula in (D, ), consider some copula E 1 ∈ D satisfying E 1 E 0 . Then E 1 ∈ E. Assuming E 1 = E 0 , the fact that κ is strictly concordance order preserving implies κ(E 1 ) < κ(E 0 ) which contradicts κ(E 0 ) = inf E∈E κ(E). So we conclude that E 1 = E 0 which implies that E 0 is a minimal copula in (D, ) and satisfies E 0 D. This proves the assertion. i.e. every copula C ∈ C is comparable to (at least) one minimal copula D ∈ m(C, ).
The following result shows that the set of minimal copulas plays a key role when searching for the minimal value of a continuous and (strictly) concordance order preserving functional. It turns out that, for any continuous and concordance order preserving map κ, the set of minimizers contains at least one minimal copula. It further turns out that any continuous map that is even strictly concordance order preserving is minimized by minimal copulas only.

Theorem.
Let κ : C → R be a continuous and concordance order preserving map. Further assume that κ is also strictly concordance order preserving. Proof. The first three assertions follow from Proposition 4.1 and Theorem 4.3. We now prove the second part. To this end, assume that κ is strictly concordance order preserving. Since m(κ, D) is non-empty, by Proposition 4.1, there exists some copula D 0 ∈ m(κ, D) satisfying κ(D 0 ) = inf D∈D κ(D). To show that D 0 is a minimal copula in (D, ), consider some copula D 1 ∈ D satisfying D 1 D 0 . Then D 1 ∈ m(κ, D). Further, assume that D 1 = D 0 . Since κ is strictly concordance order preserving we hence obtain κ(D 1 ) < κ(D 0 ) which contradicts κ(D 0 ) = inf D∈D κ(D). Therefore, D 1 = D 0 , which concludes that D 0 is a minimal copula in (D, ). This proves the assertion.
Note that the results (2.1) and (2.2) in Theorem 4.5 can not be extended to concordance order preserving functionals that fail to be strictly concordance order preserving since a copula attaining the minimal value of such a functional may fail to be a minimal copula; see, e.g., Corollary 5.3 together with Example 3.10. Further note that, for d ≥ 3, the inclusion in Theorem 4.5 (2.2) is strict, in general; see, e.g., Example 5.4.

Measures of Concordance
In the following we apply the results of the previous sections to measures of concordance: First of all, we show that in the class of all continuous and concordance order preserving measures of concordance Kendall's tau is particular since it is minimized by every minimal copula. As a consequence of Theorem 4.5, it further turns out that every continuous and strictly concordance order preserving measure of concordance is minimized by minimal copulas only. Since the latter result is applicable to Spearman's rho we may conclude that every copula minimizing Spearman's rho is also a minimizer of Kendall's tau.
We employ the quite general definition of a measure of concordance proposed in [15]; compare also [7; 36; 37]: A map κ : C → R is said to be a measure of concordance if it satisfies the following axioms: The identity κ(γ(C)) = κ(C) holds for all γ ∈ Γ π and all C ∈ C.

Example. (Kendall's tau)
The map κ (τ ) : C → R given by is a continuous and concordance order preserving measure of concordance, and is called Kendall's tau; this definition of Kendall's tau is in accordance with that in [25]. κ is a continuous and strictly concordance order preserving measure of concordance; the definition of Spearman's rho used here is in accordance with that in [25]. Even though κ (ρ) can be minimized only by minimal copulas (Theorem 4.5), its minimal value is known only for d ∈ {2, 3}: In the case d = 2, W is the only copula minimizing Spearman's rho with It immediately follows from Theorem 4.5 and Theorem 3.9 that every continuous and strictly concordance order preserving measure of concordance is minimized by minimal copulas only, and that every minimal copula minimizes Kendall's tau: hold for every continuous and strictly concordance order preserving measure of concordance κ. Indeed, for d = 3, consider the minimal copula ν 1 (M) and the minimal copula C discussed in Example 3.3 (3). Then, by [27,Example 7], we obtain For d ≥ 4, the minimal copulas ν 1 (M) and ν 1,2 (M) satisfy This proves the assertion.
The relationship between measures of concordance, in particular between bivariate Kendall's tau and bivariate Spearman's rho, has received considerable attention in literature; see, e.g., [2; 12; 19; 33]. We are able to contribute to this topic by showing that every minimizer of Spearman's rho is also a minimizer of Kendall's tau: For more details on shuffles of copulas we refer to [8]. The copulas A and B satisfy A(u) ≤ B(u) for all u ∈ I 2 and hence A B with A = B and, by [16,Corollary 5.2], we obtain Moreover, for d ≥ 3, define the functions C, D : I 2 × I d−2 → I by letting By [34, Theorem 6.6.3], C and D are copulas, C D with C = D, and, by [16,Corollary 5.2], we obtain Thus, Kendall's tau is not strictly concordance order preserving.
Note that, for d ≥ 4, the non-strictness of Kendall's tau also follows from Theorem 4.5 and Example 3.10.

A Appendix
In this section we prove Theorem 3.9; the idea of the proof is as follows: For a copula C ∈ C\C τ −CM , we construct a copula D ∈ C satisfying D C with D = C which then implies that C is not a minimal copula. More precisely, we extract some comonotonic part of the given copula C and construct a related copula D in which this comonotonic part is "made non-comonotonic".
The following result provides the basis for the construction of the copula D: For every copula C ∈ C\C τ −CM there exist some p ∈ (0, 0.5] and some Proof. Consider C ∈ C\C τ −CM . Then, by Proposition 3.7, there exists some u ∈ (0, 1) 1]]. Then p ∈ (0, 0.5]. Since every copula is continuous, the map I → I given by α → C(αu) is continuous as well. Thus, there exists some β ∈ (0, 1] satisfying βu ∈ (0, 1), βu ≤ u and C(βu) = p. This proves the assertion. 1]] for some p ∈ (0, 0.5] and some a, b ∈ (0, 1) with a ≤ b, we first define the maps C (a) , C (b) : I d → I by letting Then C (a) and C (b) are d-dimensional distribution functions on I d . We further define the maps C (1,a,b) , C (2,a,b) : I d → I by letting and Then C (1,a,b) and C (2,a,b) are also d-dimensional distribution functions on I d .
(1) The marginal distribution functions of C (1,a,b) and C (2,a,b) are identical.
Assertion (1) is immediate from the definition and implies that the map C − 2p C (1,a,b) + 2p C (2,a,b) has uniform margins. Now, we prove (2). To this end, consider u, v ∈ I d with u ≤ v. Since C is a copula and a ≤ b, we first obtain K⊆{1,...,d} where the last inequality follows from the fact that C (2,a,b) is a distribution function. It remains to show that the identity C − 2p C (1,a,b) + 2p C (2,a,b) (η k (u, 0)) = 0 holds for every k ∈ {1, ..., d} and every u ∈ I d , but this follows from the fact C is a copula and C (1,a,b) and C (2,a,b) are distribution functions on I d .
Motivated by Lemma A.2, we now put D := C −2p C (1,a,b) +2p C (2,a,b) and show that D C with D = C.
-First, assume that u 1 < b 1 . We then have  -Now, assume that u 1 ≥ b 1 ≥ a 1 . We then have -If u i < b i for some i ∈ {2, ..., d}, then we obtain -If u i ≥ b i ≥ a i for every i ∈ {2, ..., d}, then we obtain  for all u ∈ I d . This proves (3), and (4) is a consequence of (1), (2) and (3).