J. Math. Anal. Appl. 417 (2014) 451–468

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Best-possible bounds on the set of copulas with given degree of non-exchangeability G. Beliakov a , B. De Baets b , H. De Meyer c,∗ , R.B. Nelsen d , M. Úbeda-Flores e a

School of IT, Deakin University, Burwood 3125, Australia Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B-9000 Gent, Belgium c Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281 S9, B-9000 Gent, Belgium d Department of Mathematical Sciences, Lewis & Clark College, 0615 S.W. Palatine Hill Rd., Portland, OR 97219, USA e Department of Mathematics, University of Almería, Carretera de Sacramento s/n, 04120 La Cañada de San Urbano, 04120 Almería, Spain b

a r t i c l e

i n f o

Article history: Received 17 September 2013 Available online 20 February 2014 Submitted by U. Stadtmueller Keywords: Best-possible bounds Copula Lipschitz condition Measure of non-exchangeability Quasi-copula

a b s t r a c t We establish best-possible bounds on the set of quasi-copulas with given degree of non-exchangeability. These bounds are shown to be best-possible bounds as well for the set of copulas with given degree of non-exchangeability, and, consequently, also on the set of bivariate distribution functions of continuous random variables with given margins and given degree of non-exchangeability. Non-exchangeability of a (quasi-)copula is measured in the sense of Nelsen, i.e. proportional to the maximal absolute difference between this (quasi-)copula and its transpose. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Exchangeable random variables play an important role in many areas of statistics, such as limit laws, extreme value theory, Bayesian statistics and stochastic processes (see, for instance, [15]). Two random variables X and Y are said to be exchangeable if the random vectors (X, Y ) and (Y, X) have the same distribution. Equivalently, if H denotes the joint distribution function of X and Y , then X and Y are exchangeable if H(x, y) = H(y, x) for all x, y ∈ [−∞, ∞]. If X and Y are not exchangeable, i.e. H(x, y) = H(y, x) for some x and y, the supremum of |H(x, y) − H(y, x)| can be used to measure the non-exchangeability of X * Corresponding author. E-mail addresses: [email protected] (G. Beliakov), [email protected] (B. De Baets), [email protected] (H. De Meyer), [email protected] (R.B. Nelsen), [email protected] (M. Úbeda-Flores). http://dx.doi.org/10.1016/j.jmaa.2014.02.025 0022-247X/© 2014 Elsevier Inc. All rights reserved.

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and Y . Dividing this supremum by its maximal possible value (1/3), one obtains the following degree of non-exchangeability taking values in the unit interval [0, 1] [21,24]: μ(H) = 3 ·

sup (x,y)∈[−∞,∞]2

  H(x, y) − H(y, x).

(1)

The maximal degree of non-exchangeability in a multivariate setting was only uncovered recently [18]. A set of axioms for measures of non-exchangeability for bivariate vectors of continuous and identically distributed random variables is given in [11] (see [2,6,10,12,13,16,31] for more studies and applications). Our goal is to establish best-possible bounds on the set of bivariate distribution functions of continuous random variables with given margins and given degree of non-exchangeability. Due to Sklar’s theorem, it suffices to study this problem for bivariate distribution functions with margins that are uniform on [0, 1], widely known as bivariate copulas. A (bivariate) copula is a function C : [0, 1]2 → [0, 1] that satisfies: (C1) boundary conditions: for any t ∈ [0, 1], it holds that C(t, 0) = C(0, t) = 0 and C(t, 1) = C(1, t) = t; (C2) 2-increasingness: for any u1 , u2 , v1 , v2 ∈ [0, 1] such that u1  u2 and v1  v2 , it holds that   VC [u1 , u2 ] × [v1 , v2 ] := C(u2 , v2 ) − C(u2 , v1 ) − C(u1 , v2 ) + C(u1 , v1 )  0. VC ([u1 , u2 ] × [v1 , v2 ]) is called the C-volume 1 of the rectangle [u1 , u2 ] × [v1 , v2 ]. Sklar’s theorem [32] states that for any bivariate distribution function H with margins F and G, there exists a copula C (which is uniquely determined on Range(F ) × Range(G)) such that H(x, y) = C(F (x), G(y)) for any x, y ∈ [−∞, ∞]. See [23] for an extensive review of this concept. If F and G are continuous, then the copula C is uniquely determined. In that case, the degree of non-exchangeability of H (Eq. (1)) coincides with the degree of non-exchangeability of the corresponding copula C, which can be written as: μ(C) = 3 ·

max

(u,v)∈[0,1]2

  C(u, v) − C(v, u).

(2)

Let C ∗ denote the transpose of a copula C, given by C ∗ (u, v) = C(v, u) for any u, v ∈ [0, 1], then (2) can be further rewritten as μ(C) = 3 max |C − C ∗ |. In this paper, we will rely on an important family of copulas, called the shuffles of M (see [22,23]). These copulas have their mass uniformly spread on line segments (with slope +1 or −1) in the unit square. The concept of a quasi-copula is a more general notion than that of a copula, and was introduced by Alsina et al. [1] in order to characterize operations on distribution functions that can or cannot be derived from operations on random variables defined on the same probability space. A (bivariate) quasi-copula is a function Q : [0, 1]2 → [0, 1] which satisfies condition (C1), but instead of (C2), the weaker conditions [17]: (Q1) Q is increasing in each variable; (Q2) 1-Lipschitz continuity: for any u1 , u2 , v1 , v2 ∈ [0, 1], it holds that   Q(u1 , v1 ) − Q(u2 , v2 )  |u1 − u2 | + |v1 − v2 |. While every copula is a quasi-copula, there exist proper quasi-copulas, i.e. quasi-copulas that are not copulas. Distinctions concerning the mass distribution of copulas and (proper) quasi-copulas can be found in [26]. Furthermore, since quasi-copulas are a special type of aggregation function, they are becoming popular in fuzzy set theory [5,8,9,19]. The degree of non-exchangeability of a quasi-copula Q is similarly defined as μ(Q) = 3 max |Q − Q∗ | (with Q∗ the transpose of Q), and is also taking values in the unit interval [0, 1] [24]. 1

We will also compute the C-volume of functions C : [0, 1]2 → [0, 1] that are not copulas.

G. Beliakov et al. / J. Math. Anal. Appl. 417 (2014) 451–468

453

One of the most important occurrences of quasi-copulas in statistics is due to the following observation [27,30]. Every set of (quasi-)copulas has a smallest upper bound and a greatest lower bound in the set of quasi-copulas (in the sense of pointwisely ordered functions) [30]. Explicitly, for such a set S, these so-called ‘best-possible’ bounds L and U are given by       L(u, v) = inf Q(u, v)  Q ∈ S and U (u, v) = sup Q(u, v)  Q ∈ S . Of course, these bounds do not necessarily belong to the set S, nor are they necessarily copulas if the set S consists of copulas only. In particular, the set of all copulas has as lower bound W (u, v) := max(0, u + v − 1) and as upper bound M (u, v) := min(u, v), known as the Fréchet–Hoeffding bounds. The same bounds turn out to be best-possible for the set of quasi-copulas as well, i.e. for every quasi-copula Q it holds that W  Q  M . One can also restrict the attention to subclasses of copulas, for instance those coinciding on some given domain or those having the same value for a measure of association. Consider the diagonal section δ : [0, 1] → [0, 1] of Q defined by δ(t) = Q(t, t). On the set of (quasi-)copulas with given diagonal section δ, the best-possible lower bound is the Bertino copula [4], while the best-possible upper bounds on the sets of quasi-copulas and copulas do not coincide in general [34]. For related work, see [7,20,28]. More generally, Tankov [33] has established the best-possible bounds on the set of quasi-copulas that coincide on a given compact subset S of [0, 1]2 and has investigated sufficient conditions on S such that these bounds are also the best-possible bounds on the set of copulas that coincide on S. In the extreme case of the set of copulas that coincide in a single point, the latter turns out to be always the case [23]. Similarly, the best-possible bounds on the set of copulas with a given measure of association, such as Kendall’s τ or Spearman’s ρ, have been studied [25,29]. The present work seamlessly fits into the latter direction. Our paper is organized as follows. In Section 2, we establish the best-possible bounds on the set of quasi-copulas with given degree of non-exchangeability, expressed in Theorem 1, while in Section 3, we show that the lower (resp. upper) bound can be written as the minimum (resp. maximum) of a number of copulas with the same degree of non-exchangeability. The latter implies that these bounds are best-possible bounds on the set of copulas with that degree of non-exchangeability, expressed in Theorem 2. 2. Best-possible bounds on the set of quasi-copulas with given degree of non-exchangeability We first tackle the problem of establishing best-possible bounds on the set of quasi-copulas Qt with a given degree of non-exchangeability t ∈ [0, 1]. Let S t and S t denote the pointwise infimum and supremum of Qt : S t (u, v) = inf{Q(u, v) | Q ∈ Qt } and S t (u, v) = sup{Q(u, v) | Q ∈ Qt }, for any u, v ∈ [0, 1]. Explicit expressions for these best-possible bounds are given in the following theorem. Theorem 1. For any t ∈ [0, 1], the pointwise infimum S t and pointwise supremum S t of Qt are the quasicopulas given by   S t (u, v) = max 0, u + v − 1, min(u − 1 + 2t/3, v − 1 + 2t/3) ,   S t (u, v) = min u, v, max(1 − t, u − t/3, v − t/3, u + v − t) . Proof. We will deal with each of the bounds separately. Let      Ut = (u, v) ∈ [0, 1]2  (∃Q ∈ Qt ) μ(Q) = 3Q(u, v) − Q(v, u) . Obviously, if (u, v) ∈ Ut , then also (v, u) ∈ Ut . We therefore let Dt = {(u, v) ∈ Ut | u  v}.

(3) (4)

G. Beliakov et al. / J. Math. Anal. Appl. 417 (2014) 451–468

454

Fig. 1. Domain Dt1 ∪ Dt2 in which points (a, b) can be situated.

Part 1: the lower bound Consider (a, b) ∈ Dt . For any quasi-copula Q ∈ Qt for which it holds that |Q(a, b) − Q(b, a)| = t/3 = T , there exists a σ  0 such that min(Q(a, b), Q(b, a)) = W (a, b)+σ and max(Q(a, b), Q(b, a)) = W (a, b)+T +σ. From max(Q(a, b), Q(b, a))  min(a, b) = b, it follows that a  1 − T − σ and b  σ + T . Furthermore, since Q is 1-Lipschitz continuous, it must hold that T  a − b. Hence, the point (a, b) lies in the triangular domain delimited by the lines u = 1 − T − σ, v = T + σ and v = u − T . Clearly, Dt is a subset of the union of all these triangular domains with σ varying from 0 to (1 − t)/2, which is exactly the triangular domain associated with σ = 0. From the constructive part of the proof below, it will follow that Dt is precisely this union. The triangular domain with corner points (2T, T ), (1 − T, T ) and (1 − T, 1 − 2T ) (which will turn out to be Dt ) can be seen as the union of two triangular subdomains Dt1 and Dt2 , respectively lying below and above the opposite diagonal of the unit square (see Fig. 1). As we are interested in constructing the pointwise infimum of all quasi-copulas that satisfy |Q(a, b) − Q(b, a)| = T for some (a, b) ∈ Dt , we only retain those that coincide with W in either (a, b) or (b, a), i.e. those for which σ = 0. On the triangular domain Δ = {(u, v) ∈ [0, 1]2 | u  v}, this pointwise infimum S t is determined by all such quasi-copulas that attain Q(a, b) = W (a, b) (and Q(b, a) = W (a, b) + T ). As this pointwise infimum will obviously be symmetric, it is then immediately determined on [0, 1]2 . Consider (a, b) ∈ Dt1 ∪ Dt2 . We will construct a quasi-copula Q(a,b) such that Q(a,b) (a, b) = W (a, b) and Q(a,b) (b, a) = W (a, b) + T , and that is, moreover, minimal on Δ. We will do so by first constructing the smallest 1-Lipschitz continuous increasing function Q∗ on Δ such that Q∗ (a, b) = W (a, b) and Q∗ (b, a) = W (a, b) + T . Obviously, it should hold that Q∗ (a, a) = W (a, b) + T and hence Q∗ (τ, a) = W (a, b) + T , for any τ ∈ [b, a]. The 1-Lipschitz continuity then implies that the diagonal section δ ∗ of Q∗ on [b, a] is given by δ ∗ (τ ) = W (a, b) + T + τ − a. On Δ, the values of Q∗ can be obtained using a construction due to Beliakov [3]:   Q∗ (u, v) := max δ ∗ (τ ) − (τ − u)+ + (τ − v)+ , τ ∈[b,a]

where z + = max(0, z). As it should hold that W  Q(a,b) , we then find that on Δ, Q(a,b) is given by  

Q(a,b) (u, v) := max W (u, v), max δ ∗ (τ ) − (τ − u)+ + (τ − v)+ , τ ∈[b,a]

which can be rewritten as

G. Beliakov et al. / J. Math. Anal. Appl. 417 (2014) 451–468

455

 

Q(a,b) (u, v) := max W (u, v), max W (a, b) + T + τ − a − (τ − u)+ + (τ − v)+ . τ ∈[b,a]

Given the diagonal section δ(u) := Q(a,b) (u, u) on [0, 1] resulting from this construction, there are various ways to complete Q(a,b) on [0, 1]2 \ Δ such that Q(a,b) (b, a) = W (a, b) + T . As we are only interested in Q(a,b) on Δ, we refrain from giving a full expression. The successful construction of Q(a,b) implies that (a, b) ∈ Dt , as announced earlier, i.e. Dt = Dt1 ∪ Dt2 . Hence, for any (u, v) ∈ Δ, the lower bound S t (u, v) is given by S t (u, v) =

min Q(a,b) (u, v)

(a,b)∈Dt

= max W (u, v), min

 

max W (a, b) + T + τ − a − (τ − u)+ + (τ − v)+ .

(a,b)∈Dt τ ∈[b,a]

Some algebraic manipulations lead to   max W (a, b) + T + τ − a − (τ − u)+ + (τ − v)+

τ ∈[b,a]

= max W (a, b) + T + τ − a − max(τ − u, τ − v, 2τ − u − v, 0) τ ∈[b,a]

 = max min W (a, b) + T + τ − a − τ + u, W (a, b) + T + τ − a − τ + v, τ ∈[b,a]

 W (a, b) + T + τ − a − 2τ + u + v, W (a, b) + T + τ − a   = min W (a, b) + T + u − a, W (a, b) + T + v − a, W (a, b) + T − a + A(a,b) (u, v)   = min W (a, b) + T + v − a, W (a, b) + T − a + A(a,b) (u, v) with A(a,b) (u, v) := max min(u + v − τ, τ ). τ ∈[b,a]

It is easy to verify that

A(a,b) (u, v) =

⎧ ⎪ ⎨ a, ⎪ ⎩

if a  if b 

u+v 2 ,

u + v − b,

if

u+v 2

u+v 2 , u+v 2 

a,

 b.

Hence,

S t (u, v) = max W (u, v), min min W (a, b) + T + v − a, min W (a, b) + T − a + A(a,b) (u, v) . (a,b)∈Dt

(a,b)∈Dt

We now compute each of the internal minima separately. Firstly, min W (a, b) + T + v − a = min

(a,b)∈Dt





min 1 T + v − a, min 2 b + T + v − 1

(a,b)∈Dt

(a,b)∈Dt

= min(v + 2T − 1, v + 2T − 1) = v + 2T − 1. For the computation of the second minimum, depending on the value of u+v 2 , we need to subdivide the domain Dt into one, two or three subdomains, which we denote as I, II and III, respectively. They are defined by (see Fig. 2):

G. Beliakov et al. / J. Math. Anal. Appl. 417 (2014) 451–468

456

Fig. 2. Generic subdivision of Dt into three subdomains I, II, III. u+v • I = {(x, y) ∈ Dt | x  u+v 2 ∧ y  2 }; u+v • II = {(x, y) ∈ Dt | x  2 ∧ y  u+v 2 }; u+v u+v • III = {(x, y) ∈ Dt | x  2 ∧ y  2 }.

Each of these subdomains can have a non-empty intersection with the opposite diagonal depending on u+v the value of u+v fixed and under the 2 . We now compute the minimum on these subdomains keeping 2 assumption that the subdomains are non-empty. (i) Minimum on subdomain I: min W (a, b) + T − a + A(a,b) (u, v) = min W (a, b) + T

(a,b)∈I

(a,b)∈I



= min

min

(a,b)∈I,a+b1

T,

min

(a,b)∈I,a+b1

a+b−1+T

= min(T, T ) = T. (ii) Minimum on subdomain II: min W (a, b) + T − a + A(a,b) (u, v)

(a,b)∈II

= min W (a, b) + T − a + (a,b)∈II

u+v 2

 u+v u+v , min b−1+T + 2 (a,b)∈II,a+b1 2 (a,b)∈II,a+b1   u+v u+v u+v = min T − (1 − T ) + ,T − 1 + T + = 2T − 1 + . 2 2 2 

= min

min

T −a+

(iii) Minimum on subdomain III: min W (a, b) + T − a + A(a,b) (u, v)

(a,b)∈III

= min W (a, b) + T − a + u + v − b (a,b)∈III



G. Beliakov et al. / J. Math. Anal. Appl. 417 (2014) 451–468

= min

min

(a,b)∈III,a+b1

T − a − b + u + v,

min

(a,b)∈III,a+b1

457

T +u+v−1



= min(T − 1 + u + v, T + u + v − 1) = T + u + v − 1. Given the above information, we are now able to compute min W (a, b) + T − a + A(a,b) (u, v).

(a,b)∈Dt

We distinguish the following two cases: 0  T  1/4 and 1/4  T  1/3. (i) If 0  T  1/4, then 0  T  2T  1 − 2T  1 − T and we consider five possible intervals belong to. (a) If u+v 2  1 − T , then only subdomain I is not empty, whence

u+v 2

can

min W (a, b) + T − a + A(a,b) (u, v) = T.

(a,b)∈Dt

(b) If 1 − 2T 

u+v 2

 1 − T , then subdomains I and II are not empty, whence 

min W (a, b) + T − a + A

(a,b)∈Dt

(c) If 2T 

u+v 2

(a,b)

u+v (u, v) = min T, 2T − 1 + 2

 = 2T − 1 +

u+v . 2

 1 − 2T , then subdomains I, II and III are not empty, whence

  u+v ,T + u + v − 1 min W (a, b) + T − a + A(a,b) (u, v) = min T, 2T − 1 + 2 (a,b)∈Dt = 2T − 1 + (d) If T 

u+v 2

u+v . 2

 2T , then subdomains II and III are not empty, whence

  u+v min W (a, b) + T − a + A(a,b) (u, v) = min 2T − 1 + ,T + u + v − 1 2 (a,b)∈Dt = 2T − 1 + (e) If 0 

u+v 2

u+v . 2

 T , then subdomain type III is not empty, whence min W (a, b) + T − a + A(a,b) (u, v) = T + u + v − 1.

(a,b)∈Dt

Summarizing, for 0  T  1/4, it holds that ⎧ ⎪ ⎨ T + u + v − 1, (a,b) min W (a, b) + T − a + A (u, v) = 2T − 1 + u+v 2 , ⎪ (a,b)∈Dt ⎩ T,

u+v 2  T, u+v T  2 1− 1 − T  u+v 2 .

if 0  if if

T,

(ii) If 1/4  T  1/3, then 0  T  1 − 2T  2T  1 − T and we again need to consider five intervals can belong to. This analysis leads to the same result as in (i).

u+v 2

458

G. Beliakov et al. / J. Math. Anal. Appl. 417 (2014) 451–468

Summarizing, for any (u, v) ∈ Δ, the lower bound S t is given by    S t (u, v) = max W (u, v), min 2T + v − 1, B(u, v) , with ⎧ ⎪ ⎨ T + u + v − 1, B(u, v) = 2T − 1 + u+v 2 , ⎪ ⎩ T,

u+v 2  T, T  u+v 2 1− 1 − T  u+v 2 .

if 0  if if

T,

We show that the contribution of B(u, v) in the expression for S t (u, v) is irrelevant. (i) If 0  u+v 2  T , then B(u, v) = T + u + v − 1  3T − 1  0  W (u, v). u+v (ii) If T  2  1 − T , then B(u, v) = 2T − 1 + u+v 2  2T + v − 1. (iii) Finally, consider 1 − T  u+v . If B(u, v) < v − 1 + 2T , then v > 1 − T and W (u, v)  u + v − 1  2 2v − 1 > 1 − 2T  T = B(u, v). Since S t is symmetric, it now follows that for any (u, v) ∈ [0, 1]2 , the lower bound S t is given by   S t (u, v) = max 0, u + v − 1, min(u − 1 + 2T, v − 1 + 2T ) . Part 2: the upper bound In Part 1, we have proven that all points (a, b) ∈ Δ for which a quasi-copula Q ∈ Qt exists such that |Q(a, b) − Q(b, a)| = T , are situated in Dt . For any (a, b) ∈ Dt , we can find a quasi-copula Q(a,b) ∈ Qt such that min(Q(a,b) (a, b), Q(a,b) (b, a)) = M (a, b)−T = b−T and max(Q(a,b) (a, b), Q(a,b) (b, a)) = M (a, b) = b. Indeed, such a quasi-copula Q(a,b) exists if and only if min(Q(a,b) (a, b), Q(a,b) (b, a))  W (a, b), or, equivalently, b − T  max(a + b − 1, 0), and the latter inequality is trivially fulfilled for all (a, b) ∈ Dt . On the triangular domain Δ = {(u, v) ∈ [0, 1]2 | u  v}, S t can be found as the pointwise supremum of all quasi-copulas Q ∈ Qt for which Q(a, b) = M (a, b) and Q(b, a) = M (a, b) − T . As S t is obviously symmetric, it is then immediately determined on [0, 1]2 . Consider again (a, b) ∈ Dt . We will construct a quasi-copula Q(a,b) such that Q(a,b) (a, b) = M (a, b) = b and Q(a,b) (b, a) = M (a, b) − T = b − T , and that is, moreover, maximal on Δ. We will do so by first constructing the greatest 1-Lipschitz continuous increasing function Q∗ on Δ such that Q∗ (a, b) = b and Q∗ (b, a) = b − T . Obviously, it should hold that Q∗ (b, b) = b − T and hence Q∗ (b, τ ) = b − T , for any τ ∈ [b, a]. The 1-Lipschitz continuity then implies that the diagonal section δ ∗ of Q∗ on [b, a] is given by δ ∗ (τ ) = τ − T. On Δ, the values of Q∗ can be obtained using again the construction due to Beliakov [3]:   Q∗ (u, v) := min δ(τ ) + (u − τ )+ + (v − τ )+ . τ ∈[b,a]

As it should hold that Q(a,b)  M , we then find that on Δ, Q(a,b) is given by 

 Q(a,b) (u, v) := min M (u, v), min τ − T + (u − τ )+ + (v − τ )+ . τ ∈[b,a]

Given the diagonal section δ(u) := Q(a,b) (u, u) on [0, 1] resulting from this construction, there are various ways to complete Q(a,b) on [0, 1]2 \ Δ such that Q(a,b) (b, a) = b − T . As we are only interested in Q(a,b) on Δ, we refrain from giving a full expression. Hence, for any (u, v) ∈ Δ, the upper bound S t is given by

G. Beliakov et al. / J. Math. Anal. Appl. 417 (2014) 451–468

459

S t (u, v) = max Q(a,b) (u, v) (a,b)∈Dt

= min v, max

 

min τ − T + (u − τ )+ + (v − τ )+ .

(a,b)∈Dt τ ∈[b,a]

Some algebraic manipulations lead to   min τ − T + (u − τ )+ + (v − τ )+ = min τ − T + max(u − τ, v − τ, u + v − 2τ, 0)

τ ∈[b,a]

τ ∈[b,a]

= min max(u − T, v − T, u + v − T − τ, τ − T ) τ ∈[b,a]

= min max(u − T, u + v − T − τ, τ − T ) τ ∈[b,a]

  = max u − T, D(a,b) (u, v) − T with D(a,b) (u, v) := min max(u + v − τ, τ ). τ ∈[b,a]

It is easy to verify that ⎧ u+v ⎪ ⎨ u + v − a, if a  2 , D(a,b) (u, v) = u+v if b  u+v 2 , 2  a, ⎪ ⎩ u+v b, if 2  b. Hence,

S t (u, v) = min v, max u − T, max D(a,b) (u, v) − T . (a,b)∈Dt

We now compute the internal maximum. As in Part 1, we need to consider separately the three subdomains of Dt , denoted before as I, II, and III (see Fig. 2). We compute the maximum on these subdomains keeping u+v 2 fixed and under the assumption that the subdomains are non-empty. (i) Maximum on subdomain I: max D(a,b) (u, v) − T = max u + v − a − T = u + v − 3T.

(a,b)∈I

(a,b)∈I

(ii) Maximum on subdomain II: max D(a,b) (u, v) − T = max

(a,b)∈II

(a,b)∈II

u+v u+v −T = − T. 2 2

(iii) Maximum on subdomain III: max D(a,b) (u, v) − T = max b − T = 1 − 3T.

(a,b)∈III

(a,b)∈III

Given the above information, we are now able to compute max D(a,b) (u, v) − T.

(a,b)∈Dt

We distinguish again the two cases: 0  T  1/4 and 1/4  T  1/3.

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(i) If 0  T  1/4, then 0  T  2T  1 − 2T  1 − T and we consider five possible intervals belong to. (a) If u+v 2  1 − T , then only subdomain I is not empty, whence

u+v 2

can

(ii) If 1/4  T  1/3, then 0  T  1 − 2T  2T  1 − T and we again need to consider five intervals can belong to. This analysis leads to the following result:

u+v 2

max D(a,b) (u, v) − T = u + v − 3T.

(a,b)∈Dt

(b) If 1 − 2T 

u+v 2

 1 − T , then subdomains I and II are not empty, whence 

max D

(a,b)

(a,b)∈Dt

(c) If 2T 

u+v 2

u+v (u, v) − T = max u + v − 3T, −T 2

 = u + v − 3T.

 1 − 2T , then subdomains I, II and III are not empty, whence   u+v − T, 1 − 3T max D(a,b) (u, v) − T = max u + v − 3T, 2 (a,b)∈Dt  1 1 − 3T, if 2T  u+v 2  2, = u + v − 3T, if 12  u+v 2  1 − 2T.

(d) If T 

u+v 2

 2T , then subdomains II and III are not empty, whence 

max D

(a,b)∈Dt

(e) If 0 

u+v 2

(a,b)

u+v (u, v) − T = max − T, 1 − 3T 2

 = 1 − 3T.

 T , then subdomain type III is not empty, whence max D(a,b) (u, v) − T = 1 − 3T.

(a,b)∈Dt

Summarizing, for 0  T  1/4, it holds that  max D(a,b) (u, v) − T =

(a,b)∈Dt

1 − 3T,

if 0 

u + v − 3T, if

⎧ ⎪ ⎨ 1 − 3T, (a,b) max D (u, v) − T = u+v 2 − T, ⎪ (a,b)∈Dt ⎩ u + v − 3T,

if 0 

1 2



u+v 2 u+v 2

u+v 2

 12 ,  1.

 1 − 2T,

if 1 − 2T 

u+v 2

if 2T 

 1.

u+v 2

 2T,

Summarizing, for any (u, v) ∈ Δ, the upper bound S t is given by    S t (u, v) = min v, max u − T, E(u, v) , with ⎧ ⎪ ⎨ 1 − 3T, E(u, v) = u+v 2 − T, ⎪ ⎩ u + v − 3T,

if 0  if if

u+v 1 2  min( 2 , 1 − 2T ), 1 min( 12 , 1 − 2T )  u+v 2  max( 2 , 2T ), max( 12 , 2T )  u+v 2  1.

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Fig. 3. Support and values of the copula S t given by (3).

We show that the contribution of E(u, v) in the expression for S t (u, v) is only partly relevant. The inequality u+v 2 − T  u − T can only be satisfied on Δ if u = v, in which case both sides equal u − T . We obtain   max u − T, E(u, v) = max(u − T, u + v − 3T, 1 − 3T ). Since S t is symmetric, it now follows that for any (u, v) ∈ [0, 1]2 , the upper bound S t is given by   S t (u, v) = min u, v, max(u − T, v − T, u + v − 3T, 1 − 3T ) .

2

3. Copula decomposition of the lower and upper bound By construction, S t and S t are quasi-copulas. Before proceeding with the main result of this paper, we will further investigate the structure of these quasi-copulas. In view of the proof of the main theorem, we show that S t (resp. S t ) can be decomposed (by means of the minimum (resp. maximum) operation) into copulas that have a degree of non-exchangeability equal to t. Proposition 1. For the lower bound function S t given by (3), it holds that: (a) S t is a copula for any t ∈ [0, 1]. For any t ∈ [0, 3/4], S t = W , and for any t ∈ ]3/4, 1], S t is a shuffle of M with t-dependent support shown in Fig. 3. ∗ (b) S t = min(C1,t , C1,t ), with C1,t given by   C1,t (u, v) = max 0, u + v − 1, min(t/3, u − 1 + 2t/3, v, u + v − 1 + t/3) .

(5)

(c) C1,t is a copula, in particular a shuffle of M with support shown in Fig. 4(a). (d) C1,t ∈ Qt , or, equivalently, μ(C1,t ) = t. (e) For any t  t ∈ [0, 1], it holds that S t  S t . Proof. (a) For any (u, v) ∈ Δ, it holds that S t (u, v) = max(0, u + v − 1, v − 1 + 2t/3). If u + v  1, then it holds that S t (u, v) = max(0, v − 1 + 2t/3). Since v  12 , it then follows that S t (u, v) = 0 for any t ∈ [0, 3/4]. If u + v  1, then it holds that S t (u, v) = max(u + v − 1, v − 1 + 2t/3). This maximum equals u + v − 1 if u + v − 1  v − 1 + 2t/3 or, equivalently, if u  2t/3. Since u  1/2, it follows that S t (u, v) = u + v − 1 for any

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∗ Fig. 4. Supports and values of the copulas (a) C1,t , and (b) C1,t in Proposition 1.

t ∈ [0, 3/4]. Combining these results and taking into account the symmetry of S t , it follows that S t = W for any t ∈ [0, 3/4], while for any t ∈ ]3/4, 1] the expression for S t matches the expression associated with the shuffle of M that has the support shown in Fig. 3. Note that the probability mass is uniformly spread along the following three segments: from u = 0 to u = 1 − 2t/3 and from u = 2t/3 to u = 1 on the opposite diagonal and from u = 1 − 2t/3 to u = 2t/3 on the main diagonal. Also note that S t given by (3) belongs to the Bertino family of copulas [4,14]. (b) Taking into account that for any a, b, c, d, e, f ∈ R, it holds that        min max a, min(b, c) , max a, min(d, e) = max a, min(b, c, d, e) , we need to prove that   max 0, u + v − 1, min(u − 1 + 2t/3, v − 1 + 2t/3)   = max 0, u + v − 1, min(t/3, u − 1 + 2t/3, v − 1 + 2t/3, u + v − 1 + t/3) , where we have already used that u − 1 + 2t/3 < u and v − 1 + 2t/3 < v. Since both sides are symmetric w.r.t. u and v, it suffices to analyze the situation where u  v, i.e. we must prove that   max(0, u + v − 1, v − 1 + 2t/3) = max 0, u + v − 1, min(t/3, v − 1 + 2t/3, u + v − 1 + t/3) . First, we assume that u + v  1. Then it should hold that   max(0, v − 1 + 2t/3) = max 0, min(v − 1 + 2t/3, u + v − 1 + t/3) . If v − 1 + 2t/3  0, then the equality trivially holds. If v − 1 + 2t/3 > 0, then u  v > 1 − 2t/3  t/3 and therefore u + v − 1 + t/3 > v − 1 + 2t/3. The equality again holds. Next, we assume that u + v  1. Then it should hold that   max(u + v − 1, v − 1 + 2t/3) = max u + v − 1, min(t/3, v − 1 + 2t/3) . If v − 1 + 2t/3  t/3, then the equality trivially holds. If v − 1 + 2t/3 > t/3, then u  v > 1 − t/3  2t/3, whence u + v − 1 > v − 1 + 2t/3. The equality again holds.

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Fig. 5. Support and values of the function S t given by (4) when t > 3/4 (the dashed discontinuous line joining (1 − 2t/3, 1 − 2t/3) to (2t/3, 2t/3) means negative mass). ∗ Hence, it is proven that S t = min(C1,t , C1,t ). ∗ (c) Figs. 4(a)–(b) show the supports and values of C1,t and C1,t , computed from expression (5). Clearly, C1,t is a shuffle of M with probability mass uniformly distributed on the support shown in Fig. 4(a). (d) From Fig. 4(a), or equivalently, from expression (5), we get C1,t (1 − t/3, t/3) = t/3 and C1,t (t/3, 1 − ∗ t/3) = 0. Hence μ(C1,t ) = t. Due to symmetry, also μ(C1,t ) = t. (e) Follows immediately from (3). 2

Proposition 2. For the upper bound function S t given by (4), it holds that: (a) For any t ∈ [0, 3/4], S t is a copula, while for any t ∈ ]3/4, 1], it is a proper quasi-copula. For any t ∈ [0, 1/2], S t = M , and for any t ∈ [1/2, 3/4] it is a shuffle of M with support shown in Figs. 6(a)–(c), i.e. containing three segments for t ∈ ]1/2, 3/5], five segments for t ∈ ]3/5, 3/4[ and four segments for t = 3/4. ∗ ∗ (b) S t = max(C2,t , C2,t , C3,t , C3,t ), with C2,t and C3,t respectively given by   C2,t (u, v) = min u, v, max(1 − t, u − 2t/3, v − t/3, u + v − 1) ,   C3,t (u, v) = min u, v, max(0, u − 2t/3, v − t/3, u + v − t) .

(6) (7)

(c) C2,t and C3,t are copulas, i.e. shuffles of M with support shown in Figs. 7(a)–(b), respectively. (d) C2,t ∈ Qt and C3,t ∈ Qt , or, equivalently, μ(C2,t ) = μ(C3,t ) = t. (e) For any t  t ∈ [0, 1], it holds that S t  S t . Proof. (a) First note that for any t ∈ ]3/4, 1], we have VS t ([1 − 2t/3, 2t/3]2 ) = 1 − 4t/3 < 0, whence S t is a proper quasi-copula for any such t. The t-dependent support of the proper quasi-copula is shown in Fig. 5. On the dashed segment, negative probability mass is uniformly distributed. On Δ, i.e. for u  v, it holds that S t (u, v) = min(v, max(u − t/3, u + v − t, 1 − t)). Hence, S t (u, v) can only be different from v if there exists a point (u, v) ∈ Δ such that u − t/3  v, u + v − t  v and 1 − t  v, or, equivalently, such that 1 − t  v  u  t. Clearly, for t ∈ [0, 1/2] no such point can be found. By symmetry, it follows that S t = M for any t ∈ [0, 1/2]. For any t ∈ [1/2, 3/4], it turns out that S t is a shuffle of M (different from M itself), but the relative position of the segments that constitute the support varies as t moves from 1/2 to 3/4. We need to distinguish two cases. The maximum of u − t/3, u + v − t and 1 − t equals u − t/3 in those points (u, v) ∈ Δ for

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Fig. 6. Supports and values of the copula S t for (a) t ∈ (1/2, 3/5], (b) t ∈ (3/5, 3/4) (where p = v − t/3 and q = u − t/3), and (c) t = 3/4 in Proposition 2.

which it holds that v  2t/3 and u  1 − 2t/3. Furthermore, this maximum is strictly smaller than v if and only if u − t/3 < v. This situation can only arise when there exist points (u, v) ∈ Δ such that 1 − t  u − t/3 < v  2t/3, hence only if t > 3/5. We conclude that for any t ∈ [1/2, 3/5] the expression for S t (u, v) reduces to   S t (u, v) = min u, v, max(u + v − t, 1 − t) . It follows that S t is the shuffle of M with support shown in Fig. 6(a). The expressions of S t are given for different subsets of the unit square. Finally, for any t ∈ ]3/5, 3/4], expression (4) for S t cannot be simplified further. Each of the expressions that occur as arguments of the maximum can generate the maximum value, which in turn can be smaller than the minimum of u and v. In this case, the copula S t is a shuffle of M with support shown in Fig. 6(b). Fig. 6(c) shows the situation for t = 3/4. It is the highest value of t for which S t is a copula. (b) Taking into account that for any a, b, c, d, e, f ∈ R it holds that        max min a, max(b, c) , min a, max(d, e) = min a, max(b, c, d, e) ,

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Fig. 7. Supports and values of the copulas (a) C2,t , and (b) C3,t in Proposition 2.

we need to prove that   min u, v, max(u − t/3, v − t/3, u + v − t, 1 − t)   = min u, v, max(1 − t, u − 2t/3, u − t/3, v − 2t/3, v − t/3, u + v − 1, 0, u + v − t) . Clearly, the expressions u − 2t/3, v − 2t/3, u + v − 1 and the value 0 can be removed from the argument list. ∗ ∗ It follows that the equality is then identically satisfied. Hence, S t = max(C2,t , C2,t , C3,t , C3,t ). (c) Fig. 7 shows the supports and values of C2,t and C3,t , computed from expressions (6) and (7), respectively. Clearly C2,t and C3,t are shuffles of M , whence they are copulas. (d) From expressions (6) and (7), we obtain C2,t (1 − t/3, 1 − 2t/3) = 1 − t and C2,t (1 − 2t/3, 1 − t/3) = 1 − 2t/3, whence μ(C2,t ) = t, and C3,t (2t/3, t/3) = 0 and C3,t (t/3, 2t/3) = t/3, whence μ(C3,t ) = t. (e) Follows immediately from (4). 2 Note that for any t ∈ [0, 1], S t is a copula (see Proposition 1(a)). This is in agreement with a theorem of Tankov [33] which states that for the best-possible lower bound on a set of quasi-copulas which coincide on a compact subset of [0, 1]2 to be a copula, it suffices that the subset is increasing (that is, any two points are comparable). In the present case, this subset is a closed section situated on the main diagonal of the unit square, hence a compact increasing subset. Similarly, for the best-possible upper bound on such a set of quasi-copulas to be a copula, it suffices that the subset is decreasing (that is, any two points are incomparable). This sufficient condition is not fulfilled in our case, whence it is not guaranteed that S t is always a copula, a fact that has been confirmed by Proposition 2(a). However, for any t ∈ [0, 1], it still holds that S t can be written as the pointwise maximum of four copulas, as expressed by Proposition 2(b)–(c). 4. Best-possible bounds on the set of copulas with given degree of non-exchangeability The analysis of the best-possible bounds on the set of quasi-copulas Qt with given degree of nonexchangeability t ∈ [0, 1] in Propositions 1–2 implies that these bounds are also best-possible bounds on the set of copulas St with the same degree of non-exchangeability t.

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Theorem 2. For any t ∈ [0, 1] and any copula C ∈ St , it holds that St  C  St, where the bounds S t and S t are given by (3) and (4), respectively. Moreover these bounds are best-possible bounds, i.e.    S t (u, v) = inf C(u, v)  C ∈ St ,

   S t (u, v) = sup C(u, v)  C ∈ St .

Proof. In Theorem 1, we have proven that S t and S t are the best-possible lower and upper bounds on the set Qt . Obviously, S t (resp. S t ) is also a lower (resp. upper) bound on the set St . There remains to prove that these bounds are best-possible bounds. Since the lower bound S t is the pointwise minimum of ∗ ∗ two copulas C1,t ∈ St and C1,t ∈ St , we can find for any fixed (u, v) ∈ [0, 1]2 a copula Ct (C1,t or C1,t , for example) with degree of non-exchangeability t, and such that Ct (u, v) = S t (u, v), whence the lower bound is a best-possible bound. Similarly, since S t is the pointwise maximum of four copulas in St , the upper bound is a best-possible bound too. 2 As an immediate consequence of Propositions 1(e) and 2(e), we obtain the following corollary. Corollary 1. For any t ∈ [0, 1] and any copula C ∈

 t ∈[t,1]

St , it holds that

St  C  St. The translation of Theorem 2 in terms of distribution functions immediately leads to the following corollary. Corollary 2. Let X and Y be continuous random variables with joint distribution function H and marginal distribution functions F and G, respectively. If μ(H) = t, then the best-possible bounds for H are     S t F (x), G(y)  H(x, y)  S t F (x), G(y) for all (x, y) ∈ [−∞, ∞]2 . 5. Discussion The interested reader will have observed from Propositions 1 and 2 that for a given degree of nonexchangeability t ∈ [0, 1/2], the copulas S t and S t are nothing else but the Fréchet–Hoeffding bounds, and as such do not carry interesting information. For higher t, the bounds S t and S t get closer together. More explicitly, let us define for two copulas C1 and C2 their enclosed volume as    C1 (u, v) − C2 (u, v) du dv. V (C1 , C2 ) = 6 [0,1]2

For the Fréchet–Hoeffding bounds, it holds that V (M, W ) = 1, the greatest volume possible. In particular, the volume m(t) = V (S t , S t ) is given by ⎧ 1, if t ∈ [0, 12 ], ⎪ ⎪ ⎪ ⎪ ⎨ −8t3 + 12t2 − 6t + 2, if t ∈ ] 12 , 35 ], v(t) = 34 3 14 2 ⎪ if t ∈ ] 35 , 34 ], ⎪ 27 t − 3 t + 4t, ⎪ ⎪ ⎩ 94 3 18 2 − 27 t + 3 t − 4t + 2, if t ∈ ] 34 , 1].

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With increasing degree of non-exchangeability t, the volume v(t) decreases rather slowly in the interval [1/2, 3/4] where S t = W (with m(1/2) = 1, m(3/5) = 124/125 and m(3/4) = 29/32), while it decreases more rapidly in the interval [3/4, 1], ending at m(1) = 14/27. Acknowledgments The fifth author acknowledges the support of the Ministerio de Ciencia e Innovación (Spain) and FEDER, under research project MTM2009-08724. References [1] A. Alsina, R.B. Nelsen, B. Schweizer, On the characterization of a class of binary operations on distribution functions, Statist. Probab. Lett. 17 (1993) 85–89. [2] T. Bacigál, V. Jágr, R. Mesiar, Non-exchangeable random variables, Archimax copulas and their fitting to real data, Kybernetika 47 (2011) 519–531. [3] G. Beliakov, Construction of aggregation operators for automated decision making via optimal interpolation and global optimization, J. Ind. Manag. Optim. 3 (2007) 193–208. [4] S. Bertino, Sulla dissomiglianza tra mutabili cicliche, Metron 35 (1977) 53–88. [5] T. Calvo, A. Kolesárová, M. Komorníková, R. Mesiar, Aggregation operators: properties, classes and construction methods, in: T. Calvo, G. Mayor, R. Mesiar (Eds.), Aggregation Operators – New Trends and Applications, Physica-Verlag, Heidelberg, 2002, pp. 3–105. [6] B. De Baets, H. De Meyer, R. Mesiar, Asymmetric semilinear copulas, Kybernetika 43 (2007) 221–233. [7] B. De Baets, H. De Meyer, M. Úbeda-Flores, Opposite diagonal sections of quasi-copulas and copulas, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 17 (2009) 481–490. [8] B. De Baets, S. Janssens, H. De Meyer, Meta-theorems on inequalities for scalar fuzzy set cardinalities, Fuzzy Sets and Systems 157 (2006) 1463–1476. [9] S. Díaz, S. Montes, B. De Baets, Transitivity bounds in additive fuzzy preference structures, IEEE Trans. Fuzzy Systems 15 (2007) 275–286. [10] F. Durante, Construction of non-exchangeable bivariate distribution functions, Statist. Papers 50 (2009) 383–391. [11] F. Durante, E.P. Klement, C. Sempi, M. Úbeda-Flores, Measures of non-exchangeability for bivariate random vectors, Statist. Papers 51 (2010) 687–699. [12] F. Durante, R. Mesiar, L∞ -measure of non-exchangeability for bivariate extreme value and Archimax copulas, J. Math. Anal. Appl. 369 (2010) 610–615. [13] F. Durante, P.-L. Papini, Non-exchangeability of negatively dependent random variables, Metrika 71 (2010) 139–149. [14] G.A. Fredricks, R.B. Nelsen, The Bertino family of copulas, in: C. Cuadras, J. Fortiana, J.A. Rodríguez-Lallena (Eds.), Distributions with Given Marginals and Statistical Modelling, Kluwer Academic Publishers, Dordrecht, 2002, pp. 81–91. [15] J. Galambos, Exchangeability, in: S. Kotz, N.L. Johnson (Eds.), Encyclopedia of Statistical Sciences, vol. 2, John Wiley & Sons, New York, 1982, pp. 573–577. [16] C. Genest, J. Nešlehová, Assessing and modeling asymmetry in bivariate continuous data, in: P. Jaworski, F. Durante, W.K. Härdle (Eds.), Copulae in Mathematical and Quantitative Finance, in: Lecture Notes in Statist., Springer, Berlin, Heidelberg, 2013, pp. 91–114. [17] C. Genest, J.J. Quesada Molina, J.A. Rodríguez Lallena, C. Sempi, A characterization of quasi-copulas, J. Multivariate Anal. 69 (1999) 193–205. [18] M. Harder, U. Stadtmüller, Maximal non-exchangeability in dimension d, J. Multivariate Anal. 124 (2014) 31–41. [19] S. Janssens, B. De Baets, H. De Meyer, Bell-type inequalities for quasi-copulas, Fuzzy Sets and Systems 148 (2004) 263–278. [20] E.P. Klement, A. Kolesárová, Extension to copulas and quasi-copulas as special 1-Lipschitz aggregation operators, Kybernetika 41 (2005) 329–348. [21] E.P. Klement, R. Mesiar, How non-symmetric can a copula be?, Comment. Math. Univ. Carolin. 47 (2006) 141–148. [22] P. Mikusiński, H. Sherwood, M.D. Taylor, Shuffles of Min, Stochastica 13 (1992) 61–74. [23] R.B. Nelsen, An Introduction to Copulas, second edition, Springer, New York, 2006. [24] R.B. Nelsen, Extremes of nonexchangeability, Statist. Papers 48 (2007) 329–336. [25] R.B. Nelsen, J.J. Quesada-Molina, J.A. Rodríguez-Lallena, M. Úbeda-Flores, Bounds on bivariate distribution functions with given margins and measures of association, Comm. Statist. Theory Methods 30 (2001) 1155–1162. [26] R.B. Nelsen, J.J. Quesada-Molina, J.A. Rodríguez-Lallena, M. Úbeda-Flores, Some new properties of quasi-copulas, in: C. Cuadras, J. Fortiana, J.A. Rodríguez-Lallena (Eds.), Distributions with Given Marginals and Statistical Modelling, Kluwer Academic Publishers, Dordrecht, 2002, pp. 187–194. [27] R.B. Nelsen, J.J. Quesada-Molina, J.A. Rodríguez-Lallena, M. Úbeda-Flores, Best-possible bounds on sets of bivariate distribution functions, J. Multivariate Anal. 90 (2004) 348–358. [28] R.B. Nelsen, J.J. Quesada-Molina, J.A. Rodríguez-Lallena, M. Úbeda-Flores, On the construction of copulas and quasicopulas with given diagonal sections, Insurance Math. Econom. 42 (2008) 473–483. [29] R.B. Nelsen, M. Úbeda-Flores, A comparison of bounds on sets of joint distribution functions derived from various measures of association, Comm. Statist. Theory Methods 33 (2004) 2299–2305.

468

G. Beliakov et al. / J. Math. Anal. Appl. 417 (2014) 451–468

[30] R.B. Nelsen, M. Úbeda-Flores, The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas, C. R. Acad. Sci. Paris Ser. I 341 (2005) 583–586. [31] K.F. Siburg, P.A. Stoimenov, Symmetry of functions and exchangeability of random variables, Statist. Papers 52 (2011) 1–15. [32] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8 (1959) 229–231. [33] P. Tankov, Improved Frechet bounds and model-free pricing of multi-asset options, J. Appl. Probab. 48 (2011) 389–403. [34] M. Úbeda-Flores, On the best-possible upper bound on sets of copulas with given diagonal sections, Soft Comput. 12 (2008) 1019–1025.