The key role of convexity in some copula constructions

We start with some binary (“outer”) copula, apply it to an arbitrary binary (“inner”) copula and its dual (the latter being transformed by some real function) and ask under which conditions the result is again a binary copula. Sufficient convexity conditions for the transformation function and for the “outer” copula (ultramodularity and Schur concavity) are given, thus generalizing the scenarios considered in Klement et al. (J Math Inequal 11(2):361–381, 2017) and Manstavičius and Bagdonas (Fuzzy Sets Syst 354:48–62, 2019). In general, these sufficient conditions are not necessary (and a counterexample is provided), but in some distinguished cases necessary and sufficient conditions can be given. Several well-known families of copulas can be obtained in this way. We also present a few extensions for special “outer”/“inner” copulas and/or transformation functions, as well as some counterexamples.

In this paper we present sufficient conditions under which a combination of an arbitrary copula and its dual by means of some transformation function on the unit interval and some "outer" copula yields again a copula. This is a significant generalization of the scenarios considered in [35] (where the transformation function coincides with the identity function) and in [48] (where was chosen as "outer" copula; see also [14,20,44]). In general, these sufficient conditions are not necessary (and a counterexample is provided), but in some special cases necessary and sufficient conditions can be given.
In our investigations some distinguished inequalities for real functions play a key role: the convexity [61] of the transformation function, on the one hand, and two variants of the convexity for the "outer" copula, on the other hand: the ultramodularity [49,50] and the Schur concavity [65] (see also [52,62]).
Ultramodular real functions can be found in several areas, and under different names. In the case of an n-dimensional domain, ultramodularity can be seen as a version of convexity: under mild regularity assumptions, the set of ultramodular functions coincides with the set of all functions which are both convex in each variable and supermodular [49]. In the special case n = 2, ultramodularity is just convexity along the main diagonal.
Ultramodular functions have been used in economics, in particular in game theory in the context of convex measure games [3], but they also have applications in multicriteria decision support systems [5]. In mathematical analysis, ultramodular functions appeared for the first time in [76] where they were simply called convex functions (now some authors use the term Wright convexity for them [60]). In statistics, ultramodular functions occur when modeling stochastic orders and positive dependence among random vectors [56,68] (in this context they are known also as directional convex functions). For more details about ultramodular real functions see [49].
Ultramodular binary copulas are fully characterized by the convexity of their horizontal and vertical sections [38,49], and they were studied recently in [35,39]. They describe the dependence structure of stochastically decreasing random vectors, and thus each ultramodular copula is negative quadrant dependent (NQD) [57].
The concepts of Schur convex functions and Schur concave functions (as their duals) were introduced in [65] as variants of the convexity of real functions (see also [62]). For example, each symmetric convex function is Schur convex (and each symmetric concave function is Schur concave). Each Schur concave copula is necessarily symmetric, and each associative copula is Schur concave [24]. Schur convex functions preserve a preorder called majorization [51] and play a role in some related inequal-ities [69]. An early application was the comparison of incomes, and they also appear in physics, chemistry, political science, engineering, and economics [52].
A simple example of a Schur convex function in several variables is the maximum. The minimum and the product of strictly positive factors are Schur concave, as well as all elementary symmetric functions (again only in the case of strictly positive components) [70,71]. In the framework of stochastics and aggregation functions [29], the variance and the standard deviation are Schur convex, and the Shannon entropy function, the Rényi entropy function, and the Gini coefficient are examples of Schur concave functions [4,28,45,58,59].

Given a binary 1-Lipschitz aggregation function
and it is also a binary 1-Lipschitz aggregation function.
Observe that, for any quasi-copula Q, 1 is not a neutral element of Q * , so the dual of a quasi-copula is never a quasi-copula (nor is the dual of a copula a copula). In this paper, the convexity [61] of real functions and two other properties related to distinguished inequalities for real functions (in our case, for binary copulas) will play a major role: the ultramodularity [49] and the Schur concavity [65].
Consider an n-dimensional cuboid A ⊆ R n and recall that a function f : If A ⊆ R n then a function f : A → R is called ultramodular [49] if its increments are monotone non-decreasing, i.e., if for all x, y ∈ A with x y and all h 0 such that Therefore, a binary copula C : [0, 1] 2 → [0, 1] is ultramodular [35,38,39] if and only if for all x, y, z ∈ [0, 1] 2 satisfying x + y + z ∈ [0, 1] 2 , Note that, as a consequence of [ [6]) we know that C is an ultramodular copula if and only if the function Considering, for example, the family of Clayton copulas [57] generated by the family (t λ : , we see that the corresponding Archimedean copulas are ultramodular if and only if λ > 0, i.e., exactly the nilpotent Clayton copulas are ultramodular.
The next distinguished property of binary copulas which will be crucial in our work, the Schur concavity [65], is a special type of monotonicity in the sense that it reverses majorization [51].
Consider a pair x = (x 1 , x 2 ) ∈ R 2 and assign to it a pair x ↓ = (x ↓ 1 , x ↓ 2 ) ∈ R 2 which has the same components, but sorted in descending order, i.e., x ↓ 1 x ↓ 2 . Given two pairs y = (y 1 , y 2 ) ∈ R 2 and x = (x 1 , x 2 ) ∈ R 2 , then y is said to majorize x (in symbols y x) if 1 x 1 + x 2 = y 1 + y 2 and y ↓ 1 x ↓ 1 . Note that is not a partial order on R 2 because it is not antisymmetric: from y x and x y we only can conclude that x and y have the same components, but not necessarily in the same order.
If A ⊆ R 2 then a function f : A → R is Schur convex [65] if f preserves majorization, i.e., y Clearly, the three basic copulas W , and M are Schur concave, as well as each associative copula. Observe that each Schur concave copula is symmetric [24].
In our context the Schur concavity of a copula D : [0, 1] 2 → [0, 1] is only required on the upper left triangle = {(x, y) ∈ [0, 1] 2 | x y}, i.e., we only need the Schur concavity of the restriction D : → [0, 1] of D to . This is equivalent with saying that for all (x, y) ∈ and for all ε > 0 with (x + ε, y − ε) ∈ , Observe that symmetric copulas which are Schur concave on are necessarily Schur concave (on the whole unit interval). However, a copula which is Schur concave on need not be symmetric: for instance, the copula D : [0, 1] 2 → [0, 1] given by is not symmetric (and, therefore, not Schur concave), but the restriction D of D to coincides with , so D is Schur concave on .

Main result
We shall deal in this paper with several classes of monotone functions from one real interval to another: where the subscript "dec" stands for "monotone non-increasing" and "conc" for "concave". We also shall work with the class of 1-Lipschitz functions from the unit interval [0, 1] into some real interval [c, d] with c 1 d: 1] . Then obviously f (1) = 1. If f is not 1-Lipschitz, i.e., there exist x 0 , y 0 ∈ ]0, 1[ with x 0 < y 0 such that f (y 0 ) = f (x 0 ) + k(y 0 − x 0 ) for some k > 1, then, because of the convexity of f , we have 1 = f (1) f To further simplify our notations, we put Note that in the special case f = id [0,1] formula (3.1) reduces to We want to know under which conditions, for each binary copula C : [0, 1] 2 → [0, 1], the composite function D(C, f (C * )) defined by (3.1) is also a binary copula. In this context, we also will refer to f as "transformation function", to D as "outer" and to C as "inner" copula.  . Observe that in this case the original result in [48] was given for functions of the form given by (3.1) is a binary copula.
As far as we know, the third of these results (for the special case D = M) is new.
is a copula. This shows that (i) implies (ii).
For the converse, assume that, for each copula C, also M(C, f (C * )) is a copula and that there is an and, in particular, E(x, x) = min(x, f (x)) for all x ∈ [0, 1].
Taking into account the continuity and the monotonicity of the copula E, f (1) = 1 implies that there is an i.e., E is not 2-increasing, which is a contradiction to our assumptions.
Our main result generalizes these results and provides sufficient conditions for the transformation function f and the outer copula D such that, for each copula C, also the function D(C, f (C * )) is a copula. Proof Fix an arbitrary binary copula C and a binary copula D which is ultramodular and Schur concave on , and write briefly We first show that the assumptions in Theorem 3.5 imply that E satisfies both boundary conditions of copulas. For each x ∈ [0, 1] we have and, as f (1) = 1, we also get Similarly, we get E(0, x) = 0 and E(1, x) = x, i.e., E is grounded with neutral element 1.
We have to show that From the monotonicity and the 1-Lipschitz property of C we get 0 α δ and 0 β ε. The 2-increasingness of C implies α + β γ , and the 1-Lipschitz property gives γ δ + ε.
Using (3.3) and (3.5), we can rewrite (3.4) as (3.6) To simplify the notation put u = C(x, y) and v = C * (x, y). Then (3.6) may be rewritten as Identifying, as usual, two-dimensional vectors with points in [0, 1] 2 and putting can be written as If we replace in (3.8)-(3.11) the values u and v by their original meaning, i.e., and take into account that for all (a, b) Consider also the point P 5 such that P 1 , P 2 , P 3 , and P 5 are the vertices of a parallelogram, i.e., (3.13) and the point P 6 given by (3.14) Then, from the convexity of f it follows that (3.15) and the 1-Lipschitz property of f (see Lemma 3.1) yields Adding the left-and right-hand sides of the inequalities (3.15) and (3.16) and canceling the expression f (v + ε − β + δ − α) on both sides of the sum, we obtain i.e., p 62 p 52 (see (3.13) and (3.14)). Since p 51 = p 61 , we have In order to complete the check that E is 2-increasing we distinguish the following three cases: Fig. 1 (left)).
Since D is ultramodular on we may apply [35, (2.7)] to the parallelogram with the vertices P 1 , P 2 , P 3 , and P 5 , and we obtain From (3.17) and the monotonicity of D we get Finally, we want to show that D(P 4 ) D(P 6 ). Clearly (see (3.11) and (3.14)) we have Note that for each point P = (x, y) on the line segment connecting P 4 and P 6 we have The slope of this line equals −1, i.e., it is orthogonal to the main diagonal of the unit square. Moreover, Hence, from the Schur concavity of D on (see [35, p. 367 showing that E is 2-increasing in this case. Case 2: P 5 ∈ and P 6 / ∈ [0, 1] 2 (as shown in Fig. 1 (right)). Consider two additional points: Q 1 is the common point of the upper border l u of the unit square (i.e., the line segment from (0, 1) to (1, 1)) and of the line connecting P 4 and P 6 . Q 2 is the intersection point of l u and the line connecting P 5 and P 6 (see Fig. 1 (right)). Clearly, we have Considering the points Q 2 and P 5 , on the one hand, and Q 1 and Q 2 , on the other hand, the monotonicity of D implies D(Q 2 ) D(P 5 ) and D(Q 1 ) D(Q 2 ). (3.21) Again, for all points P = (x, y) on the line segment connecting P 4 and P 6 (in particular, for P 4 and Q 1 ) we have q 12 ), and the Schur concavity of D on (compare [35]) allows us to conclude that D(P 4 ) D(Q 1 ).
Case 3: P 5 / ∈ [0, 1] 2 (as visualized in Fig. 2). First of all, observe that P 6 / ∈ [0, 1] 2 because of (3.17). We consider four additional points Q 3 , Q 4 , Q 5 , and Q 6 as follows: is the intersection point of the upper border l u of the unit square and the line connecting the points P 3 and P 5 .
Without going into details, we have S , there is some λ 3 > 0 such that The coordinates of Q 1 satisfy the equation q 11 − p 61 + q 12 − p 62 = 0 because the line connecting P 6 and Q 1 is orthogonal to the main diagonal of the unit square, i.e., Q 1 = (q 61 + 1 − q 62 + λ 1 + λ 2 + λ 3 , 1). Since the Fréchet-Hoeffding lower bound W is the smallest copula, i.e., D W , we have which means that E is 2-increasing also in this case, thus completing the proof.    (i) If the copula C is Schur concave on then also the copula D(C, f (C * )) is Schur concave on . (ii) If the copula C is ultramodular on then also the copula D(C, f (C * )) is ultramodular on .
Proof In order to show (i) assume that the copula C is Schur concave on , and let us briefly write, as in (3.3), E = D(C, f (C * )). Then for all (x, y) ∈ and for all α ∈ 0, min 1 − x, y, y−x 2 we obtain C(x + α, y − α) C(x, y), i.e., we have that C(x + α, y − α) = C(x, y) + β for some β 0. Hence, Now, using (3.23) together with the monotonicity of D and its Schur concavity on , (3.22) can be written as i.e., E is Schur concave on . When proving (ii), assume that the copula C is ultramodular on . Again putting E = D(C, f (C * )), it suffices to show that all horizontal and vertical sections of E are convex on .

(3.25)
Since ε γ + δ we may write ε = γ + δ + ζ for some ζ 0. Using the 1-Lipschitz − ζ , and the monotonicity of D allows us to conclude that Now the Schur concavity of D on implies for the right-hand side of (3.26)

and (3.25) can be transformed into
Further, the convexity of f yields and, because of the monotonicity of f , we may write for some a, b 0, implying Now, using (3.29), the monotonicity of D and the ultramodularity of D on , we may rewrite (3.27) as Finally, going back and using the relations in (3.28) and (3.24) together with the definition of E, we obtain i.e., the horizontal section E( · , y 0 ) is convex.
Fixing an arbitrary x 0 ∈ [0, 1], the convexity of the vertical section E(x 0 , · ) on [x 0 , 1] can be shown in complete analogy, completing the proof of (ii).
It is evident that, for a function f : [0, 1] → [0, 1], the condition f (1) = 1 is necessary for D(C, f (C * )) satisfying the boundary conditions of copulas for each copula C. In [35] it was shown that the ultramodularity of D on is a necessary condition for D(C, C * ) being a copula for each copula C. This result can be easily carried over to our scenario.

Proposition 3.9 Let D be a binary copula such that for each binary copula C and for each f ∈ F [0, 1] [0,1] the function D(C, f (C * )) is a copula. Then D is ultramodular on the upper left triangle .
Proof It is enough to consider f = id    a + (1 − a)x. (3.30) Then for the Fréchet-Hoeffding lower bound, i.e., for D = W as outer copula we have as a consequence of Theorem 3.5: (i) For each a ∈ [0, 1] and each copula C the function W (C, f a (C * )) given by is a copula (this result can already be found in [19, Lemma 3.1, (3.4)]). Observe that also the convex combination C a = aC + (1 − a)W is a copula which is given by C a (x, y) = aC(x, y) + max ((1 − a)(x + y − 1), 0). Hence W (C, f a (C * )) C a , and these two copulas coincide whenever x + y 1. (ii) In particular, for each a ∈ [0, 1], the function W ( , f a ( * )) is a binary copula. Note that the family of copulas (W ( , f a ( * ))) a∈ [0,1] is just the family of Sugeno-Weber copulas [74,75] (compare also [

Some extensions of Theorem 3.5: leaving the unit square
In this section, we will look more closely at properties of the transformation function However, this situation changes if, for instance, we fix the inner copula C (and maybe also the outer copula D) and we want to know for which transformation functions f : [0, 1] → [0, 1] the construction (3.1) leads to a copula for these special copulas C (and D).  ). These copulas were studied in [21] (based on some earlier results in [15,17] is a copula. Indeed, (C λ ) λ∈]0,1[ is a subfamily of the family of Cuadras-Augé copulas [40,57].
In some particular cases we can obtain similar results as in Theorem 3.5, although we leave the unit square [0, 1] 2 , i.e., the framework of binary copulas. It is easy to check that While, for each θ ∈ [−1, 0], we have f 1+θ ∈ F [0, 1] [0,1] (and Theorem 3.5 can be applied), for θ ∈ ]0, 1] we see that the function 2 − f 1−θ is monotone non-increasing with Ran and Theorem 3.5 does not apply. Note again that, for θ ∈ ]0, 1], the product in the composite function · (2 − f 1−θ )( * ) operates outside of [0, 1] 2 , so it is necessary to use the dot symbol rather than the copula : (a similar approach was described in [31]).
Since the function · (2 − f 1−θ )( * ) is a copula for each θ ∈ ]0, 1], this means that in this special case (where both the outer and the inner copula coincide with ) construction (3.1) leads to a copula even if the hypotheses of Theorem 3.5 are not satisfied.
Observe that, for an arbitrary copula C, the function C · (2 − f 1−θ (C * )) is well defined, but not necessarily a copula. For example, for C = M and θ = 1 we obtain the contradiction

Example 4.3
Recall the family of copulas (N C α ) α∈ [−1,1] given by To the best of our knowledge, these copulas were mentioned for the first time in [8], and they have a special invariance with respect to weighted geometric means: for all α, β ∈ [−1, 1] and all θ ∈ [0, 1] we have It is not difficult to check that, for all α ∈ [−1, 0] and for the corresponding functions , i.e., g α does not satisfy the hypotheses of Theorem 3.5, but in the case when both the outer and the inner copula coincide with construction (3.1) yields the copula · g α ( * ) = 2 −N C α given by the function q (a,c) can be written as a convex combination of q (0,0) , q (0. 5,0.5) and q (1,0) as follows: (1,0) . 1] , we may start with an outer copula D which is ultramodular and Schur concave on , use the construction (3.1), and obtain for each inner copula C the copula D(C, q (a,c) (C * )).

Concluding remarks
We have introduced and discussed a rather general construction for binary copulas, generalizing several previous approaches. To be precise, we have considered an outer copula D :  [35]) and (C, f (C * )) (i.e., where the outer copula D coincides with the product copula [48]). Moreover, we have shown that, in particular cases, it is possible to leave the unit interval [0, 1] (as far as the range of f is concerned), in which case a binary function F : [0, 1] × Ran( f ) → R should replace the outer copula D. This was exemplified in the case when F equals the product on [0, 1] × [0, 2] ⊆ R 2 and the inner copula C coincides with .
More generally, the same procedure can be applied to some outer copula D (which is ultramodular and Schur concave on ), to all affine functions f λ : [0, 1] → [0, 1] with λ ∈ [−1, 0] as given by (3.30), and to an arbitrary inner copula C. Writing As already stressed, the class of functions F [0, 1] [0,1] allows us to construct copulas based on an outer copula D (being ultramodular and Schur concave on ) and on an arbitrary inner copula C. As an interesting problem for further research one can fix an appropriate outer copula D and look for all functions f such that D(C, f (C * )) is a copula for each inner copula C. Similarly, one can fix the copulas D and C and search for feasible functions f .