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≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 3$$\end{document}, 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 monotone association between random variables 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. [1]: -the random vector has the upper Fréchet-Hoeffding bound M as its copula, -each of its coordinates is a.s. 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. [2]. In the bivariate case, the strongest notion of negative monotone association between random variables 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. [1]: -the random vector has the lower Fréchet-Hoeffding bound W as its copula, -each of its coordinates is a.s. 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][5][6][7][8].
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) 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. [9]. 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 [10] and Lee et al. [11] 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 [7,12] for further results on variance minimization. Moreover, Genest et al. [13] and Lee and Ahn [6] have provided minimal copulas minimizing certain measures of concordance. Further continuous and concordance order preserving functionals on copulas are discussed in [9,14], but also in [15][16][17][18] where the necessary properties of the functional strictly depend on the function to be integrated; for more details on this topic we refer to [17] 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 3.1), and we provide a necessary condition (Theorem 3.1) for a copula to be minimal. The sufficient conditions are related to the extreme negative dependence concept of K -countermonotonicity introduced in [6,11], 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 4.1). 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 5.2). Some auxiliary results are established in the Appendix.

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}.
Copulas For K ⊆ {1, . . . , d}, we consider the map η K : I d × I d → I d given coordinatewise by and, for k ∈ {1, . . . , d}, we put η k := η {k} . A copula is a function C : I d → I satisfying the following conditions: This definition is in accordance with the literature; see, e.g. [1,19]. We denote the collection of all copulas by C. The following copulas are of particular interest: Since every copula C has a unique extension to a distribution function R d → I, there exists a unique probability measure Q C : B(I d ) → I satisfying Q C [[0, u]] = C(u) for every u ∈ I d . The probability measure Q C is said to be the copula measure with respect to C and it satisfies 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 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 ν K := k∈K ν k (such that ν ∅ = ι). 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 [20] for further details on the groups Γ π , Γ ν and Γ .
The Ordered Topological Space C Under the sup-norm, the collection C is a compact topological space.
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 order relation is said to be closed, if the graph of is a closed set in C × C. We denote by m(C, ) the set of all minimal elements of (C, ). For Then is a closed order relation which is called 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 Note that the ordered set (C, ) is not a lattice: for d = 2, this is immediate from [1, Theorem 7.3.5] and, for d ≥ 3, the assertion follows using the same arguments as in the proof of [1,Theorem 7.3.5]. In this context, recall that, for d = 2, concordance order and pointwise order coincide.
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: Remark 3.1 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 [21][22][23].
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 -a family {g k } k∈K of strictly increasing and continuous functions I → R and -some c ∈ R such that We denote by C K −CM the collection of all copulas that are K -CM. To ease notation, we write d-CM in the case K = {1, . . . , d}. Note that, for d = 2, a copula C is 2-CM if and only if C = W . Thus, K -countermonotonicity may be regarded as a natural extension of countermonotonicity to dimensions d ≥ 3. For some particular choices K ⊆ {1, . . . , d}, K -countermonotonicity implies minimality: Proof The inclusions were proved in [11,Theorem 4,Lemma 5], and the equivalence for d = 2 follows from Example 3.2 (1) below.

Examples 3.2 below show that
(1) the inclusion in Proposition 3.1 for |K | = d is strict whenever d ≥ 3, (2) the inclusion in Proposition 3.1 for |K | = d − 1 is strict, and that (3) the inclusion in Proposition 3.1 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: The assertion then follows from Proposition 3.1.
(2) The copula C given by is Archimedean and is called the Clayton copula with parameter −1/(d − 1). It follows from [6, Theorem 4] and Proposition 3.1 that C is d-CM and hence minimal.
Then, by [26, Theorem 6.6.3], D is a d-dimensional copula and it follows from Proposition 3.1 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 3.1 (2) is d-CM and hence minimal. However, it is evident that C fails to be K -CM whenever 2 ≤ |K | ≤ d − 1.
|K |-CM copula C and define the maps D, E :  (1) and (2), We proceed with the discussion of a necessary condition for a copula to be minimal. A copula C ∈ C is said to be Kendall-countermonotonic (Kendall-CM), if the identity min C(u), (τ (C))(1 − u) = 0 holds for every u ∈ ]0, 1[. We denote by C Kendall-CM the collection of all copulas that are Kendall-CM. The term Kendall-countermonotonicity is motivated by the fact that a copula C is Kendall-CM if and only if C minimizes multivariate Kendall's tau; see [5]. Kendall's tau is a map κ : C → R given by and the definition of Kendall's tau is in accordance with [27]. The following characterization of Kendall-countermonotonicity is due to [5, Lemma 3.1, Theorem 3.4]: Proposition 3.2 For a copula C ∈ C the following are equivalent: It follows from [5, Theorem 3.3] that, for d = 2, a copula C is Kendall-CM if and only if C = W . Thus, Kendall-countermonotonicity may be regarded as a natural extension of countermonotonicity to dimensions d ≥ 3.

Remark 3.3 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 A ⊆ supp Q C , i.e. the support supp Q C of Q C contains some strictly comonotonic subset of ]0, 1[. It then follows from Proposition 3.2 (3) that such a copula C fails to be Kendallcountermonotonic.
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 Kendall-CM and hence every minimal copula minimizes Kendall's tau. The following example shows that the inclusion in Theorem 3.1 is strict whenever d ≥ 4: It is interesting to note that also every K -CM copula is Kendall-CM: Proof The inclusions follow from [6, Theorem 6], and the equivalence for d = 2 is a consequence of Example 3.4 below.
The following example shows that the inclusions in Proposition 3.3 are strict whenever d ≥ 3: In the following, we summarize the obtained inclusions: where, due to Example 3.4, 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 φ : 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→∞ φ(C n ) = φ(C). concordance order preserving if the inequality φ(C) ≤ φ(D) holds for all C, D ∈ C satisfying C D. strictly concordance order preserving if it is concordance order preserving and the strict inequality φ(C) < φ(D) holds for all C, D ∈ C satisfying C D with C = D.
For a given map φ : C → R and a subset D ⊆ C, we define the set m(φ, D) by letting It is well known that every continuous functional C → R is minimized by some copula C ∈ C; see Proposition B.1: 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: is a continuous and strictly concordance order preserving functional, and thus, due to Proposition 4.1, φ is minimized by some copula C ∈ C. In [7, Corollary 4.1 and Figure  3.2] the authors have determined the value of inf C∈C φ(C) in arbitrary dimension and, for d = 3, they have presented a minimal copula minimizing φ.
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 φ, 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.  Further assume that φ is also strictly concordance order preserving. We note that the results (2.1) and (2.2) in Theorem 4.1 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 5.1 together with Example 3.3.

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 4.1, 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 first recall the definition of Kendall's tau and then introduce Spearman's rho.

Example 5.1 (Kendall's tau)
The map κ : 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 [27].
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 [27]. Even though ρ can be minimized only by minimal copulas (Theorem 4.1), 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 ρ(W ) = −1 and, for d = 3, Nelsen and Úbeda-Flores [25,Theorem 4] have shown that min C∈C ρ(C) = −1/2 and that this minimal value is attained only by those minimal copulas satisfying Q[{u ∈ I 3 : 3 i=1 u i = 3/2}] = 1.
It immediately follows from Theorem 4.1 and Theorem 3.1 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: (2) If d = 2, then the identities m(φ, C) = m(C, ) = m(κ, C) hold for every continuous and strictly concordance order preserving measure of concordance φ.
It follows from Example 3.3 that the second inclusion in Corollary 5.1 (1) is strict whenever d ≥ 4. The next example shows that the first inclusion in Corollary 5.1 (1) is strict whenever d ≥ 3: 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. [37][38][39][40]. 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 [41]. The copulas A and B satisfy A(u) ≤ B(u) for all u ∈ I 2 and hence A B with A = B and, by [5,Corollary 5.2], we obtain κ(A) = κ(B). Moreover, for d ≥ 3, define the functions C, D : [26,Theorem 6.6.3], C and D are copulas, C D with C = D, and, by [5,Corollary 5.2], we obtain κ(C) = κ(D). 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 4.1 and Example 3.3.
[b, 1] . 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 C (1,a,b) C (2,a,b) Then C (1,a,b) and C (2,a,b) are also d-dimensional distribution functions on I d .

Lemma A.2 (1)
The marginal distribution functions of C (1,a,b) and C (2,a,b) are identical.
Proof Assertion (1) is immediate from the definition and implies that the map C − 2 p C (1,a,b) +2 p 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 − 2 p C (1,a,b) + 2 p 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 − 2 p C (1,a,b) + 2 p C (2,a,b) and show that D C with D = C. C (1,a,b) (u) and D(u) ≤ C(u) hold for every u ∈ I d .

Appendix B: Some Topological Results
In this section, we recall some useful topological results. We start with the extreme value theorem: Proposition B.2 Consider a non-empty and compact topological space X equipped with a closed order . Then, for every x ∈ X , there exists some minimal element y ∈ X satisfying y x.
Applying the above propositions, we obtain the following result: Theorem B.1 Consider a non-empty and compact topological space X equipped with a closed order and let φ be a real-valued and continuous mapping on X .
(1) If φ is monotone, then the set of all minimizers of φ is non-empty and contains at least one minimal element of X . (2) If φ is strictly monotone, then every minimizer of φ is a minimal element of X .
Proof Assertion (1) is immediate from Propositions B.1 and B.2. We now prove (2). To this end, consider any y ∈ X satisfying φ(y) = inf x∈X φ(x) and any z ∈ X satisfying z y. Since φ is monotone, we obtain φ(z) = φ(y), and now the strict monotonicity of φ yields z = y. Therefore, y is a minimal element of X . This proves (2) and hence the assertion.