On the lower bound of Spearman’s footrule

Abstract Úbeda-Flores showed that the range of multivariate Spearman’s footrule for copulas of dimension d ≥ 2 is contained in the interval [−1/d, 1], that the upper bound is attained exclusively by the upper Fréchet-Hoeffding bound, and that the lower bound is sharp in the case where d = 2. The present paper provides characterizations of the copulas attaining the lower bound of multivariate Spearman’s footrule in terms of the copula measure but also via the copula’s diagonal section.


Introduction
Spearman's footrule is a "measure of disarray" (Diaconis and Graham [1]) which assigns to every copula C : [ , ] d → [ , ] a value ϕ(C) in the real numbers. Due to its "simplicity, robustness and natural interpretation" (Genest et al. [11]), it has already been used in several and quite di erent areas of research: This includes, for instance, genomics [12], information science [13] and economics [15]; we also refer to the impressive number of applications listed in [11] and [16].
Úbeda-Flores [18] showed that the range of Spearman's footrule for copulas of dimension d ≥ is contained in the interval [− /d, ], that the upper bound is attained exclusively by the upper Fréchet-Hoe ding bound, and that the lower bound is sharp in the case where d = .
In the present paper, we show that, for every dimension d ≥ , the lower bound of Spearman's footrule is sharp and attained by uncountably many copulas. To this end, we rst provide characterizations of the copulas attaining the lower bound of Spearman's footrule. It turns out that a copula minimizes Spearman's footrule if and only if the support of the corresponding copula measure is contained in the complement of the set , d− d d ∪ d , d . We then construct a sparse structure and show that every sparse copula having this sparse structure minimizes Spearman's footrule. This then leads to uncountably many copulas attaining the lower bound. The paper is organized as follows: Section 2 collects some de nitions and results on copulas and related topics which will be needed in this paper. Section 3 gives a brief discussion of Spearman's footrule and provides several characterizations of the copulas attaining the lower bound; it turns out that it is important where the mass of the copula is distributed, not how. In Section 4 we then outline the concept of sparse copulas and provide a class of sparse copulas minimizing Spearman's footrule in any dimension.

Preliminaries
Let I := [ , ] and let d ≥ be an integer which will be kept x throughout this paper. We denote by the vector in R d with all coordinates being equal to and by the vector in R d with all coordinates being equal to . In the sequel, we denote by C the collection of all d-dimensional copulas; for details on copulas we refer to [4,14]. Since every copula C ∈ C is continuous and 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 corresponding to C.
For i, j ∈ { , ..., d} with i < j, we de ne the maps π i,j : C → C by letting Each of these permutations is an involution. For the sake of simplicity, we further de ne the mapτ : I d → I d and the map τ : C → C by letting We note that the total re ection τ transforms every copula into its survival copula. We refer to [6,10] for further details on permutations and the total re ection.
The total re ection also provides an order relation on C which is called the concordance order on C

Spearman's Footrule
In this section we provide characterizations of the copulas attaining the lower bound of multivariates Spearman's footrule. The key for dealing with Spearman's footrule of a copula turns out to be the copula's diagonal section.
For a map C : I d → I, the map δ C : I → I given by is called the diagonal section of C. The following result may be found in [4, Theorem 2.6.1]:

Proposition. A function δ : I → I is the diagonal section of a copula C ∈ C if and only if it satis es the following properties
The diagonal section δ W of the d-dimensional lower Fréchet-Hoe ding bound W satis es and δ W ≤ δ C for every d-dimensional copula C. Although W fails to be a copula whenever d ≥ , due to Proposition 3.1, for every dimension d ≥ there exists some copula C satisfying δ C = δ W .
The map ϕ : C → R given by is called Spearman's footrule; here, λ denotes the restriction of the Lebesgue measure to the Borel σ-eld B(I) on I. This de nition of Spearman's footrule is in accordance with the de nitions used in [11,18]. It is immediate from [7] and [18] that ϕ is symmetric, i.e. the identity ϕ(π i,j (C)) = ϕ(C) holds for every C ∈ C and all i, j ∈ { , ..., d} with i < j; radially symmetric, i.e. the identity ϕ(τ(C)) = ϕ(C) holds for every C ∈ C; concordance order preserving, i.e. the inequality κ(C) ≤ κ(D) holds for all C, D ∈ C satisfying C D; continuous, i.e. for any sequence {Cn} n∈N ⊆ C and any C ∈ C, uniform convergence limn→∞ Cn = C implies limn→∞ κ(Cn) = κ(C); and satis es ϕ(Π) = , where Π denotes the product copula. Note that ϕ is not a measure of concordance in the sense of [2,8,17]. For more details on Spearman's footrule we refer to [11,18] and the references there.
Úbeda-Flores [18] showed that the inequalities In the sequel, we rst characterize the class of all those copulas satisfying ϕ(C) = − d in terms of their copula measure but also via the copula's diagonal section. In Section 4, we then show that this class is non-empty and contains uncountably many copulas. This implies that the above lower bound of Spearman's footrule is sharp and is attained by uncountably many copulas.

Theorem.
For a copula C ∈ C the following statements are equivalent: Proof. The equivalence of (a) and (b) is straightforward.
case, Lipschitz continuity of a copula's diagonal section (Proposition 3.1 (iv)) yields, for every u ∈ d− d , , and hence δ C (u) = du − (d − ) = δ W (u) for every u ∈ d− d , , and the same argument applies to δ τ(C) in the place of δ C . This proves the equivalence of (b) and (c). Note, that condition (c) in Theorem 3.2 solely depends on where the mass of the copula is distributed, not how. We illustrate this result for the cases d ∈ { , }:

Examples.
(1) In the bivariate case, the mass of a copula C minimizing Spearman's footrule needs to lie in the upper left and lower right rectangles below (grey area) (2) In the trivariate case, the support of the corresponding copula measure Q C needs to be a subset of the below grey subcubes In the next section we discuss a copula construction principle that ts quite well with condition (c) in Theorem 3.2. In passing, we prove the existence of a copula C minimizing Spearman's footrule and show that Spearman's footrule is minimized by uncountably many copulas.

Sparse Copulas minimizing Spearman's Footrule
In this section we outline the concept of sparse copulas. In particular, we construct a certain sparse structure and show that every sparse copula having this sparse structure minimizes Spearman's footrule.
Consider an index set J ⊆ N such that |J| ≥ . A family {[an , bn]} n∈J of non-empty closed intervals in I d is called a sparse structure, if it satis es the following conditions: (i) b n,i − a n,i = b n,j − a n,j holds for every n ∈ J and all i, j ∈ { , ..., d}.
(ii) (a m,i , b m,i ) ∩ (a n,i , b n,i ) = ∅ holds for all m, n ∈ J such that m ≠ n and for every i ∈ { , . . . , d}. It turns out that the total re ection of any sparse copula is again a sparse copula: Proof. For every u ∈ I d we rst have (τ • ψ S n )(u) = − a n, + (b n, −a n, ) u , . . . , a n,d + (b n,d −a n,d ) u d = − a n, − (b n, −a n, ) u , . . . , − a n,d − (b n,d −a for all u ∈ I d . This proves the assertion.
Any sparse copula having the below sparse structure minimizes Spearman's footrule. Thus, again, for minimizing Spearman's footrule it is important where the mass of the copula is distributed, not how. Finally, we apply Theorem 4.3 to the cases d ∈ { , , }. In particular, we show that any sparse copula of dimension d = having the same sparse structure than the lower Fréchet-Hoe ding bound W is a minimizer of Spearman's footrule:

Examples.
(1) In the case d = , the only sparse structure satisfying the assumptions of Theorem 4.3 is the sparse (3) In the case d = , there exist exactly twelve possible sparse structures satisfying the assumptions of Theorem 4.3 The above example (1) ensures that even in the case d = there exist uncountably many copulas minimizing Spearman's footrule.

Remark.
The sparse copulas provided in Theorem 4.3 all minimize Spearman's footrule. However, obviously, there also exist other copulas satisfying condition (c) in Theorem 3.2. Further minimizers of Spearman's footrule may be obtained applying sparse structures with a ner mesh J satisfying b n, − a n, = /|J|, but also those copulas induced by certain generalized transformation matrices studied in [5,Section 5]. In contrast to sparse copulas, the latter construction principle allows distributing some mass to each single subcube in the complement of the set , d− d d ∪ d , d , however, the mass assigned to each of these subcubes has to be distributed in exactly the same way.