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The Frank inequality
Accepted Manuscript The Frank inequality
B. De Baets, H. De Meyer
PII: DOI: Reference:
S0165-0114(17)30136-7 http://dx.doi.org/10.1016/j.fss.2017.03.017 FSS 7195
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Fuzzy Sets and Systems
Received date: Revised date: Accepted date:
3 November 2016 20 March 2017 28 March 2017
Please cite this article in press as: B. De Baets, H. De Meyer, The Frank inequality, Fuzzy Sets Syst. (2017), http://dx.doi.org/10.1016/j.fss.2017.03.017
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The Frank inequality B. De Baets a and H. De Meyer b a Department
of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B-9000 Gent, Belgium [email protected]
b Department
of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281 S9, B-9000 Gent, Belgium [email protected]
Abstract We investigate a functional inequality for copulas that has emerged from our study of the comparison of a set of random variables pairwisely coupled by a same copula. Any copula satisfying this inequality is necessarily symmetric and radially symmetric. Moreover, any associative copula satisfying this inequality is a solution to the well-known Frank equation. For this reason, the inequality is coined the Frank inequality. We fully characterize the associative copulas that satisfy the Frank inequality: they turn out to be either Frank copulas or ordinal sums of a same Frank copula with equidistant idempotent elements. As a by-product, we observe that Frank copulas are super-additive on the unit square. Key words: Associative copula, Frank copula, Frank equation, Frank inequality, Super-additivity, Radial symmetry.
1
Introduction
Triangular norms (t-norms for short), originating from the theory of probabilistic metric spaces, have been playing a key role in fuzzy set theory for decades. They have quickly become the unquestioned standard for modelling the pointwise intersection of fuzzy sets as well as for the modelling the conjunction in fuzzy logic [27]. The search for t-norms satisfying additional interesting properties has led to a number of interesting functional equations (see, e.g., [2,3,7,9]), resulting in well-known parametric families of triangular norms and conorms [30]. Also the theory of probabilistic metric spaces has been a source of interesting functional equations. In the past, many investigations were aimed at finding Preprint submitted to Elsevier Science
March 31, 2017
the solution(s) to functional equations in the space of uniform distribution functions [1]. A prominent example is the work of Frank [22,23] on a family of binary operations on the space of probability distribution functions which are related to sums of dependent random variables and are induced by copulas. This led him to the functional equation of simultaneous associativity and to the discovery of a one-parameter family of associative copulas which have turned out to be of great importance in certain areas of statistics [24,25,26,32,33,34]. This equation, however, when restricted to the class of associative copulas, does not uniquely characterize the Frank family of copulas, as it also has as solutions all ordinal sums of Frank copulas. Note that Frank copulas and ordinal sums thereof are more often regarded as tnorms [30] and that in this context the Frank equation has also been studied for the more general classes of uninorms and nullnorms [8]. The Frank t-norm family appears abundantly in fuzzy set theory, a prominent example being fuzzy preference modelling [15,20,21,36]. Recently, we have investigated transitivity properties of certain types of reciprocal relations that are generated by pairwisely comparing (in a probabilistic manner) random variables constituting a random vector. We have studied a variety of couplings between these random variables and our analysis has revealed that the reciprocal relations generated by many of these couplings possess transitivity properties that can be nicely characterized [11,16,17,18,19]. In case the random variables are pairwisely coupled by a same copula [11], the transitivity of the generated reciprocal relation can be captured in the general cycle-transitivity framework [10,13] only if the copula fulfills a countably infinite number of conditions, parametrized by an integer parameter (k ≥ 2), each of them taking the form of a functional inequality. Recently, we have shown that all Frank copulas satisfy each and every of these inequalities [12], and that the fulfillment of the inequality with parameter value k implies that of the inequality with parameter value k , for any 2 ≤ k < k. Realizing in this paper that the fulfillment of the inequality with k = 2 (and hence also with k > 2) trivially implies that of the Frank equation as well [11], inspired us to call this inequality the Frank inequality. In the present paper, we fully characterize the associative copulas that satisfy this inequality. Obviously, they must constitute a particular class of ordinal sums of Frank copulas. The outline of this paper is as follows. In Section 2, we recall what is known about Frank’s functional equation. In Section 3, we prove that every Frank copula is super-additive. Next, in Section 4, we introduce the Frank inequality and recall (while providing an alternative elegant proof) that all Frank copulas fulfill this inequality; moreover, we show that the Frank inequality implies both symmetry and radial symmetry. In Section 5, we fully characterize the class of associative copulas that satisfy the Frank inequality. In the final section, we explore some weaker forms of the Frank inequality and point to some families of non-associative copulas that are conjectured to satisfy the Frank 2
inequality.
2
Copulas and the functional equation of Frank
Copulas have become a basic tool in probability theory and in mathematical statistics. Let X and Y be continuous random variables, defined on a common probability space and taking values in R, with continuous distributions FX and FY and joint distribution HXY , i.e., FX (u) = Prob(X ≤ u) ,
FY (v) = Prob(Y ≤ v) ,
HXY (u, v) = Prob(X ≤ u, Y ≤ v) . Then there exists a unique copula CXY such that HXY (u, v) = CXY (FX (u), FY (v)) . Generally, a function C : [0, 1]2 → [0, 1] is called a copula if it satisfies the boundary conditions C(t, 0) = C(0, t) = 0 and C(t, 1) = C(1, t) = t for all t ∈ [0, 1], and the property of 2-increasingness, i.e., VC ([x1 , y1 ] × [x2 , y2 ]) := C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≥ 0 for all 0 ≤ x1 ≤ x2 ≤ 1 and 0 ≤ y1 ≤ y2 ≤ 1. VC ([x1 , y1 ] × [x2 , y2 ]) is called the C-volume of the rectangle [x1 , y1 ] × [x2 , y2 ]. Note that any copula C is increasing and 1-Lipschitz continuous. Two random variables X and Y are exchangeable if and only if the associated copula C is symmetric, i.e., C(x, y) = C(y, x) for all (x, y) ∈ [0, 1]2 . Any associative copula is symmetric and thus a 1-Lipschitz continuous t-norm. Conversely, any t-norm that is 2-increasing is an associative copula. For more information, we refer to [5]. The dual of a copula C is the function C˜ on [0, 1]2 defined by ˜ y) = x + y − C(x, y) , C(x, whereas the co-copula of C is the function C ∗ on [0, 1]2 defined by C ∗ (x, y) = 1 − C(1 − x, 1 − y) . On the other hand, the function Cˆ on [0, 1]2 defined by ˆ y) = x + y − 1 + C(1 − x, 1 − y) , C(x, 3
is a copula, which is called the survival copula associated to C. Its probabilistic interpretation is given by Prob(X > u, Y > v) = CˆXY (Prob(X > u), Prob(Y > v)) . For continuous random variables X and Y , the identity CXY = CˆXY characterizes the property of radial symmetry (see [35]). Note that CˆXY ≤ CXY implies CXY = CˆXY . In the study of the associativity of certain binary operations on probability distributions [22,23] and in a number of other settings as well, one encounters the functional equation T (x, y) + S(x, y) = x + y
(1)
for any (x, y) ∈ [0, 1]2 , where T is a t-norm and S a t-conorm. A couple (T, S) satisfying (1), with T a t-norm and S a t-conorm, is called a solution to (1). Since S is increasing, T is 1-Lipschitz continuous, whence T must be a copula, ˜ Thus, the problem is to determine all solutions (C, C) ˜ say C, and S must be C. for which C is an associative copula and its dual C˜ is also (simultaneously) associative. To begin, consider the following one-parameter family of strict t-norms (we use the parametrization commonly used in the framework of copulas):
1 (e−αx − 1)(e−αy − 1) Cα (x, y) = − log 1 + , α e−α − 1
α = 0 ,
(2)
and its limits C−∞ = W ,
C0 = Π ,
C+∞ = M ,
with W (x, y) = max(x + y − 1, 0), Π(x, y) = xy and M (x, y) = min(x, y). A straightforward calculation yields x + y − Cα (x, y) = 1 − Cα (1 − x, 1 − y), i.e., C˜α = Cα∗ , and since the associativity of Cα implies that of Cα∗ , it follows ¯ = R∪{−∞, +∞}, the couple (Cα , C ∗ ) is a solution to (1). that, for any α ∈ R α The remarkable and surprising fact is that these couples (Cα , Cα∗ ) and couples ˜ for which C is an ordinal sum of Cα ’s are the only solutions of (1). (C, C) Theorem 2.1 [23] Consider a t-norm T and a t-conorm S. Then the couple (T, S) is a solution to (1) if and only if T is an ordinal sum of Frank t-norms and S = T˜. Note that if (T, S) is a solution to (1) for a proper ordinal sum T , then S = T ∗ if and only if the set of idempotent elements of T is symmetric with respect to the point x = 1/2. In the language of copulas, the latter is usually presented as follows. 4
Theorem 2.2 [31] An associative copula C is radially symmetric (i.e., it ˆ if and only if it is a Frank copula or a symmetric ordinal holds that C = C) sum of Frank copulas, i.e., an ordinal sum of the form C = ( ak , bk , Cαk )k∈K , where for each k ∈ K, there exists a k ∈ K such that αk = αk and ak + bk = bk + ak = 1.
3
Super-additivity of Frank copulas
The family of Frank t-norms has been widely studied, especially in the fuzzy set community. Regarded as a family of copulas, the Frank copulas have been extensively discussed in statistics [24,28,29,32]. They have many interesting properties, among which the property of super-additivity that we will use in the proof of our main theorem. We are not aware that this property has been emphasized before in the literature. We present it here as a lemma. Lemma 3.1 Any Frank copula Cα is super-additive on [0, 1]2 , i.e., for any x1 , x2 , y1 , y2 ∈ [0, 1] such that x1 + x2 ≤ 1 and y1 + y2 ≤ 1, it holds that Cα (x1 + x2 , y1 + y2 ) ≥ Cα (x1 , y1 ) + Cα (x2 , y2 ) .
(3)
¯ For i = 1, 2, we introduce Proof Consider a Frank copula Cα , with α ∈ R. the notations e−αxi − 1 e−αyi − 1 , vi = −α , ui = −α e −1 e −1 and t = e−α − 1 . Then, ui , vi ∈ [0, 1], while t > 0 if α < 0 and −1 < t < 0 if α > 0. Furthermore, t → −1 if α → +∞, t → +∞ if α → −∞, and t → 0 if α → 0. For i = 2, Cα (xi , yi ) can then be expressed as Cα (xi , yi ) = −
1 ln(1 + tui vi ) , α
while Cα (x1 + x2 , y1 + y2 ) = −
1 ln[1 + t(u1 + u2 + tu1 u2 )(v1 + v2 + tv1 v2 )] . α
Substituting these expressions in (3), it follows that we must prove that the inequality t2 u1 u2 v1 v2 + t[u1 u2 (v1 + v2 ) + v1 v2 (u1 + u2 ) − u1 u2 v1 v2 ] + u1 v2 + u2 v1 ≥ 0 (4) holds for any u1 , u2 , v1 , v2 ∈ [0, 1] and any t ≥ −1. The l.h.s. of the latter inequality is linear in u1 . For u1 = 0, it simplifies to u2 v1 (1+v2 t), an expression 5
that is non-negative for any u2 , v1 , v2 ∈ [0, 1] and any t ≥ −1. For u1 = 1, it reduces to the expression t2 u2 v1 v2 + t[u2 (v1 + v2 ) + v1 v2 (1 + u2 ) − u2 v1 v2 ] + v2 + u2 v1 ,
(5)
which can be regarded as a linear function of u2 . For u2 = 0, the latter expression simplifies to v2 (1 + v1 t), an expression that is non-negative for any v1 , v2 ∈ [0, 1] and any t ≥ −1. For u2 = 1, it reduces to the expression (t+1)(v1 +v2 +v1 v2 t), again an expression that is non-negative for any v1 , v2 ∈ [0, 1] and any t ≥ −1. Hence, (5) is non-negative for any u2 , v1 , v2 ∈ [0, 1] and any t ≥ −1, which proves that inequality (4) holds for all u1 , u2 , v1 , v2 ∈ [0, 1] and all t ≥ −1. Since the Frank copulas are radially symmetric, the property of super-additivity can be reshaped in the following form. Lemma 3.2 For any Frank copula Cα , the inequality Cα (x1 + x2 − 1, y1 + y2 − 1) ≥ Cα (x1 , y1 ) + Cα (x2 , y2 ) − 1
(6)
holds for any x1 , x2 , y1 , y2 ∈ [0, 1] such that x1 + x2 ≥ 1 and y1 + y2 ≥ 1. ¯ Its radial symmetry is Proof Consider a Frank copula Cα , with α ∈ R. expressed by the equality Cα (x, y) = Cα (1 − x, 1 − y) + x + y − 1 , which holds for any (x, y) ∈ [0, 1]2 . When applied to inequality (3), the latter can be brought into the equivalent form Cα (1 − x1 − x2 , 1 − y1 − y2 ) + x1 + x2 + y1 + y2 − 1 ≥ Cα (1 − x1 , 1 − y1 ) + Cα (1 − x2 , 1 − y2 ) + x1 + y1 + x2 + y2 − 2 , or Cα (1 − x1 − x2 , 1 − y1 − y2 ) ≥ Cα (1 − x1 , 1 − y1 ) + Cα (1 − x2 , 1 − y2 ) − 1 . If we replace 1 − xi by xi and 1 − yi by yi (i = 1, 2), then x1 + x2 ≥ 1 and y1 + y2 ≥ 1, which proves that under the conditions stated, inequality (6) is indeed satisfied.
4
The Frank inequality
An intuitively appealing way of comparing two random variables is to consider the probability that the first one takes a value greater than or equal to 6
the second one. More formally, given a random vector (X1 , X2 , . . . , Xn ), we consider the [0, 1]-valued relation Q defined by Q(Xi , Xj ) = Prob{Xi > Xj } +
1 Prob{Xi = Xj } . 2
This relation is reciprocal, i.e., Q(Xi , Xj ) + Q(Xj , Xi ) = 1 for all i, j ∈ {1, . . . , n}. In [16,17,18,19], we studied the transitivity of this reciprocal relation in the special case that the coupling between any two components of the random vector is modelled by a same copula C, where C is either W , Π or M . In [11], we further pointed out that for a general copula C the transitivity of the reciprocal relation Q can only be captured in the cycle-transitivity framework [13,14] provided the copula C satisfies a countably infinite number of inequalities. More explicitly, we require C to satisfy the following family of inequalities: i
C(xi , yi ) − ⎛
≤ C ⎝yk +
xk +
j
j
C(xk−2j−1 , yk−2j−2 ) −
j
C(xk−2j , yk−2j−1 ) −
j
C(xk−2j−2 , yk−2j−1 ) −
j
C(xk−2j−1 , yk−2j ),
j
C(xk−2j−1 , yk−2j )
⎞
C(xk−2j , yk−2j−1 )⎠
(7)
for an arbitrary integer k > 1 and for all 0 ≤ x1 ≤ x2 ≤ · · · ≤ xk ≤ 1 and 0 ≤ y1 ≤ y2 ≤ · · · ≤ yk ≤ 1, and where the sums extend over all integer values that lead to meaningful indices of x and y. We refer to the first (i.e., k = 2) and simplest inequality of this family as the Frank inequality because, as we will prove later on, the only associative copulas that satisfy this inequality are the Frank copulas and very specific ordinal sums of Frank copulas. Note that if a copula C satisfies the Frank inequality, this means that the C-volume of any rectangle [x1 , x2 ] × [y1 , y2 ] in the unit square is bounded from above by the C-volume of the rectangle [0, x ] × [0, y ], where x = y2 − C(x1 , y2 ) and y = x2 − C(x2 , y1 ). Definition 4.1 A copula C : [0, 1]2 → [0, 1] satisfies the Frank inequality if for any 0 ≤ x1 ≤ x2 ≤ 1 and any 0 ≤ y1 ≤ y2 ≤ 1, it holds that C(x1 , y1 )+C(x2 , y2 )−C(x1 , y2 )−C(x2 , y1 ) ≤ C(y2 −C(x1 , y2 ), x2 −C(x2 , y1 )) . (8) The following proposition shows that copulas satisfying the Frank inequality have some interesting properties. 7
Proposition 4.1 Any copula C that satisfies the Frank inequality has the following properties: (i) C is symmetric; (ii) C is radially symmetric. Proof To prove (i), we set x1 = 0 and y1 = 0 in (8). The inequality then reduces to C(x2 , y2 ) ≤ C(y2 , x2 ), from which the symmetry of C immediately follows. To prove (ii), we set x2 = y2 = 1 in (8). Using the symmetry of C, it then follows that C(x1 , y1 ) + 1 − x1 − y1 ≤ C(1 − x1 , 1 − y1 ). Replacing herein x1 by 1 − x1 and y1 by 1 − y1 , it follows that the inequality turns into an equality. Hence, C is radially symmetric. We have already proven in [11] that all Frank copulas satisfy the Frank inequality. For the sake of completeness, we provide here an alternative and short proof. Proposition 4.2 [11] The Frank copulas satisfy the Frank inequality. ¯ We use the notations introProof Consider a Frank copula Cα , with α ∈ R. duced in the proof of Lemma 3.1. Since Cα (xi , yj ) = −
1 log [1 + tui vj ] , α
e−α(x2 −Cα (x2 ,y1 )) = (1 + tu2 )/(1 + tu2 v1 ) , and e−α(y2 −Cα (x1 ,y2 )) = (1 + tv2 )/(1 + tu1 v2 ) , inequality (8) is equivalent to t(1 + tu1 v1 )(1 + tu2 v2 ) ≤ t(1 + tu2 v1 )(1 + tu1 v2 ) + t2 (1 − u1 )(1 − v1 )u2 v2 . This inequality is trivially satisfied if u2 = 0 or v2 = 0. Assume that u2 = 0 and v2 = 0, then the inequality is equivalent to
t2 u2 v2 (1 − u1 )(1 − v1 ) − 1 −
u1 u2
1−
v1 v2
≥ 0.
Since the latter inequality clearly holds for any 0 ≤ u1 ≤ u2 ≤ 1 any 0 ≤ v1 ≤ ¯ v2 ≤ 1 and any t ≥ −1, Cα satisfies (8) for any α ∈ R. 8
5
Characterization of the associative copulas that satisfy the Frank inequality
In this section, similar to the original Frank equation, we restrict our attention to associative operations. When restricting to associative copulas, Theorem 2.2, Proposition 4.1 and Proposition 4.2 already indicate that the solution set consists of the Frank copulas, possibly complemented with certain symmetric ordinal sums of Frank copulas. The following theorem, the main result of this paper, identifies the structure of these ordinal sum solutions, and thus lays bare the full solution set. Theorem 5.1 An associative copula C satisfies the Frank inequality if and only if it is a Frank copula or an ordinal sum of the form C = ( i/k, (i + 1)/k, Cα )i∈I where I = {0, 1, . . . , k − 1}, for some integer k > 1, and Cα is a Frank copula. Proof As it is already known that Frank copulas are solutions, we can restrict our attention to symmetric ordinal sums of Frank copulas. Part 1. Suppose that C is such an ordinal sum that satisfies the Frank inequality. We first prove that the idempotent elements of C are equally spaced. If there is a single idempotent element u, then it must equal 12 , since 1−u is an idempotent element as well. If there are two idempotent elements u < v, then C(u, u) = C(u, v) = u and C(v, v) = v. Setting x1 = y1 = u and x2 = y2 = v in (8) yields the condition v − u ≤ C(v − u, v − u), which proves that v − u is also an idempotent element of C. In particular, if u < 1/2 and choosing v = 1 − u, it follows that also 1 − 2u and 2u are idempotent elements of C. Proceeding in the same way, we obtain that if u is an idempotent element of C, then all elements of the form ku, with k ∈ {1, 2, . . . , 1/u }, are idempotent elements as well. If C has an idempotent element other than 0 and 1 and the number of idempotent elements is finite, then there exists a smallest idempotent element u = 0, whence 1 − u is then necessarily the largest idempotent element different from 1. It thus follows that there must exist an integer n ≥ 1 such that nu = 1 − u, hence u = 1/(n + 1), a fractional number. Consequently, the set of idempotent elements of C is given by {i/(n + 1) | i ∈ {0, 1, . . . , n + 1}}. If C has an infinite number of idempotent elements, then the compactness of [0, 1] implies that there is at least one element in [0, 1] that is an accumulation point of idempotent elements (note that this accumulation point is not necessarily an idempotent element itself). But, since differences of idempotent elements yield again idempotent elements, 0 must be an accumulation 9
point. As all multiples of idempotent elements yield idempotent elements, any x ∈ [0, 1] is an accumulation point of idempotent elements of C. Hence, if the set of idempotent elements of C is countably infinite, then this set is dense in [0, 1]. Thus, the diagonal section of C coincides with the identity function on a dense subset of [0, 1], and due to its continuity, it coincides with the identity function on [0, 1]. Since M is the only copula with the identity function as diagonal section, it holds that C = M and thus not a proper ordinal sum. We can thus safely restrict our attention to ordinal sums C of the form C = ( i/k, (i + 1)/k, Cαi )i∈I , where I = {0, 1, . . . , k − 1}. We prove that all copulas Cαi must be equal to a same Frank copula. Let us set x1 = y1 = i/k in (8) for an arbitrary but fixed i ∈ {0, 1, . . . , k − 1}, and let x2 , y2 ∈ [i/k, (i + 1)/k]. The inequality then reduces to
1 i i Cαi (kx2 − i, ky2 − i) ≤ C x2 − , y2 − k k k
=
1 Cα (kx2 − i, ky2 − i) . k 0
Hence, for all i ∈ {1, 2, . . . , k − 1}, it holds that C α i ≤ C α0 . Let us further set x2 = y2 = (i + 1)/k in (8) for arbitrary but fixed i ∈ {0, . . . , k − 1}, and let x1 , y1 ∈ [0, 1/k]. The inequality now reduces to
i+1 1 i+1 i+1 Cα (kx1 , ky1 ) + − x1 − y1 ≤ C − x1 , − y1 k 0 k k k =
1 i + Cαi (1 − kx1 , 1 − ky1 ) . k k
(9)
In particular, we obtain for i = 0 that Cα0 (kx1 , ky1 ) + 1 − kx1 − ky1 ≤ Cα0 (1 − kx1 , 1 − ky1 ) , from which it follows that Cα0 is a radially symmetric associative copula. Since Cα0 cannot have idempotent elements other than 0 and 1, Cα0 must be a Frank copula different from M , i.e., α0 = +∞). Due to the radial symmetry of Cα0 , inequality (9) can be rewritten as Cα0 (1 − kx1 , 1 − ky1 ) ≤ Cαi (1 − kx1 , 1 − ky1 ) . Hence, it must hold for all i ∈ {1, 2, . . . , k − 1} that Cα 0 ≤ C α i . 10
It follows that Cαi = Cα for all i ∈ {0, 1, . . . , k − 1}. Hence, we have shown that the ordinal sum solutions are of the type mentioned in the theorem. Part 2. Conversely, we prove that any ordinal sum of the mentioned type satisfies the Frank inequality. Consider a copula C of the form C = ( i/k, (i + 1)/k, Cα )i∈I where I = {0, 1, . . . , k − 1}, for some integer k > 1, and Cα is a Frank copula. The part of the unit square where the copula C coincides with a rescaled Frank copula Cα consists of k squares of size 1/k aligned along the diagonal. Outside these squares the copula C coincides with M . The l.h.s. of the Frank inequality represents the C-volume of an arbitrary rectangle [x1 , x2 ] × [y1 , y2 ] with corner points (x1 , y1 ), (x2 , y2 ), (x1 , y2 ) and (x2 , y1 ). We distinguish the following situations covering all possibilities: (1) The four corner points fall inside a same square. (2) Two neighbouring corner points fall in a same square, the two other corner points are outside the squares. (3) Two opposite corner points fall in different squares, the other two (opposite) corner points are outside the squares. (4) One corner point falls inside a square, the three other corner points are outside the squares. (5) None of the corner points falls inside a square. Situation 1. Let us assume that i i+1 ≤ x1 ≤ x2 ≤ , k k
i i+1 ≤ y 1 ≤ y2 ≤ , k k
for arbitrary but fixed i ∈ {0, 1, . . . , k − 1}. We must verify whether the Frank inequality 1 1 Cα (kx1 − i, ky1 − i) + Cα (kx2 − i, ky2 − i) k k 1 1 − Cα (kx1 − i, ky2 − i) − Cα (kx2 − i, ky1 − i) k k 1 1 i i ≤ C(y2 − − Cα (kx1 − i, ky2 − i), x2 − − Cα (kx2 − i, ky1 − i)) k k k k is satisfied. Both arguments of the copula C in the r.h.s. can take any value in the interval [0, 1/k]. Hence, C should be identified with the copula Cα rescaled to the first square, i.e., 11
C(y2 − =
i i 1 1 − Cα (kx1 − i, ky2 − i), x2 − − Cα (kx2 − i, ky1 − i)) k k k k
1 Cα (ky2 − i − Cα (kx1 − i, ky2 − i), kx2 − i − Cα (kx2 − i, ky1 − i)) . k
Then the above inequality reduces to the Frank inequality for the copula Cα . This inequality is indeed satisfied for Frank copulas as is stated in Proposition 4.2. Situation 2. Let us first assume that the right corner points (x2 , y1 ) and (x2 , y2 ) of the given rectangle fall inside a same square, i.e., x1 <
i i+1 ≤ x2 ≤ , k k
i+1 i ≤ y1 ≤ y2 ≤ , k k
for arbitrary but fixed i ∈ {1, 2, . . . , k − 1}. The C-volume of the rectangle [x1 , x2 ] × [y1 , y2 ] in the l.h.s. of the Frank inequality equals the C-volume of the smaller rectangle [i/k, x2 ] × [y1 , y2 ] with all its corner points situated in the same square [i/k, (i + 1)/k], [i/k, (i + 1)/k]. Making use of the result of Situation 1, the Frank inequality holds for the latter C-volume. It follows that
C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) = C(i/k, y1 ) + C(x2 , y2 ) − C(i/k, y2 ) − C(x2 , y1 ) ≤ C(x2 − C(x2 , y1 ), y2 − C(i/k, y2 )) ≤ C(x2 − C(x2 , y1 ), y2 − C(x1 , y2 )) , where the last inequality follows from the fact that x1 ≤ i/k. This proves that for this specific case, the Frank inequality is satisfied. Due to the symmetry of C, the same can be immediately concluded for rectangles with the lower corner points falling inside a same square. Let us next assume that the left corner points (x1 , y1 ) and (x1 , y2 ) fall inside a same square, i.e., i i+1 ≤ x1 ≤ ≤ x2 , k k
i i+1 ≤ y 1 ≤ y2 ≤ , k k
for arbitrary but fixed i ∈ {1, 2, . . . , k − 1}. We now have 12
C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) = C(x1 , y1 ) + C((i + 1)/k, y2 ) − C(x1 , y2 ) − C((i + 1)/k, y1 ) ≤ C((i + 1)/k − C((i + 1)/k, y1 ), y2 − C(x1 , y2 )) ≤ C(x2 − C(x2 , y1 ), y2 − C(x1 , y2 )) , where the last inequality follows from the fact that for any copula C, the function x − C(x, y) is a copula, and is thus increasing in both arguments x and y. This concludes the proof that the Frank inequality is generally satisfied for Situation 2. Situation 3. This is the situation where the opposite corner points (x1 , y1 ) and (x2 , y2 ) fall inside different squares. Clearly, the other two corner points then fall outside any square. Hence, let us assume that i i+1 ≤ x 1 , y1 ≤ , k k
j+1 j ≤ x2 , y2 ≤ , k k
for arbitrary but fixed i, j ∈ {1, 2, . . . , k − 1} and 0 ≤ i < j ≤ k − 1. It follows that j−i−1 j−i+1 ≤ x2 − y 1 ≤ , k k
j−i+1 j−i−1 ≤ y2 − x 1 ≤ . k k
The Frank inequality reduces to j i 1 1 + Cα (kx1 − i, ky1 − i) + + Cα (kx2 − j, ky2 − j) − x1 − y1 k k k k ≤ C(x2 − y1 , y2 − x1 ) . or, equivalently, to Cα (kx1 −i, ky1 −i)+Cα (kx2 −j, ky2 −j) ≤ kx1 +ky1 −i−j+kC(x2 −y1 , y2 −x1 ) . We distinguish the following subcases. j−i−1 j−i−1 j−i j−i ≤ x2 − y 1 ≤ and ≤ y 2 − x1 ≤ . k k k k In this case, the Frank inequality becomes
(a) The case
Cα (kx1 − i, ky1 − i) + Cα (kx2 − j, ky2 − j) ≤ kx1 + ky1 − 2i − 1 +Cα (kx2 − j − (ky1 − i) + 1, ky2 − j − (kx1 − i) + 1) . 13
Introducing the notations s1 = kx1 − i ,
s2 = kx2 − j ,
t1 = ky1 − i ,
t2 = ky2 − j ,
the above is equivalent to the inequality Cα (s1 , t1 ) + Cα (s2 , t2 ) ≤ s1 + t1 − 1 + Cα (s2 − t1 + 1, t2 − s1 + 1) , or, since Cα is radially symmetric, the above is also equivalent to Cα (s2 , t2 ) + Cα (1 − s1 , 1 − t1 ) ≤ Cα (s2 − t1 + 1, t2 − s1 + 1) , which should be satisfied for all 0 ≤ s2 ≤ t1 ≤ 1 and all 0 ≤ t2 ≤ s1 ≤ 1. Due to the super-additivity property of the Frank copula Cα (see Lemma 3.1), this inequality is indeed satisfied. j−i j−i j−i+1 j−i+1 ≤ x 2 − y1 ≤ and ≤ y2 − x1 ≤ . k k k k Now, the Frank inequality reduces to
(b) The case
Cα (kx1 − i, ky1 − i) + Cα (kx2 − j, ky2 − j) ≤ kx1 − i + ky1 − i + Cα (kx2 − j − (ky1 − i), ky2 − j − (kx1 − i)) . Introducing the notations s1 = kx1 − i ,
s2 = kx2 − j ,
t1 = ky1 − i ,
t2 = ky2 − j ,
the above is equivalent to the inequality Cα (s1 , t1 ) + Cα (s2 , t2 ) ≤ s1 + t1 + Cα (s2 − t1 , t2 − s1 ) , which should be fulfilled for all 0 ≤ t1 ≤ s2 ≤ 1 and for all 0 ≤ s1 ≤ t2 ≤ 1. Making use of the radial symmetry of Cα , this inequality can be rewritten as Cα (s2 , t2 ) + Cα (1 − t1 , 1 − s1 ) ≤ 1 + Cα (s2 + (1 − t1 ) − 1, t2 + (1 − s1 ) − 1) , which should be satisfied for any s2 + 1 − t1 ≥ 1 and any t2 + 1 − s1 ≥ 1. Again, due to the super-additivity property of the Frank copula Cα (see Lemma 3.1), this inequality is indeed satisfied. j−i−1 j−i j−i+1 j−i ≤ x2 − y 1 ≤ and ≤ y 2 − x1 ≤ k k k k Now, the Frank inequality reduces to
(c) The case
Cα (kx1 − i, ky1 − i) + Cα (kx2 − j, ky2 − j) ≤ kx1 − i + kx2 − j , which is trivially fulfilled for any copula.
14
j−i j−i+1 j−i j−i−1 ≤ x 2 − y1 ≤ and ≤ y 2 − x1 ≤ k k k k As before, in this case the Frank inequality is trivially satisfied.
(d) The case
Situation 4. Repeating the arguments by which Situation 2 was essentially reduced to Situation 1, the present situation can be similarly reduced either to Situation 1 or Situation 3. The proof is therefore immediate. Situation 5. Depending on the location of the corner points, we can rely either on the results obtained for Situation 1 or Situation 3, or we can make use of the fact that the C-volume of the given rectangle is zero (that is, when the rectangle does not enclose any of the squares centered along the diagonal), in which case the Frank inequality is trivially satisfied. Note that all Frank copulas are Archimedean copulas, but ordinal sums of Frank copulas are not. Therefore, Theorem 5.1 yields the following corollary. Corollary 5.1 The Frank copulas are the only Archimedean copulas that satisfy the Frank inequality.
6
Discussion
Leaving aside the solutions that are proper ordinal sums, the only associative copulas that satisfy the Frank inequality are the Frank copulas. As the Frank inequality therefore appears to be tight, it is worthwhile to investigate weaker forms of it. One such form is obtained by setting the outer copula in the r.h.s. equal to the greatest copula M . The inequality then becomes C(x1 , y1 )+C(x2 , y2 )−C(x1 , y2 )−C(x2 , y1 ) ≤ min (x2 − C(x2 , y1 ), y2 − C(x1 , y2 )) , which, given a copula C, should be satisfied for all 0 ≤ x1 ≤ x2 ≤ 1 and 0 ≤ y1 ≤ y2 ≤ 1. However, it is easily verified that any copula C satisfies this inequality. Indeed, the inequality C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≤ x2 − C(x2 , y1 ) , is equivalent to C(x1 , y1 ) − C(x1 , y2 ) ≤ x2 − C(x2 , y2 ) , 15
which is satisfied for any copula C, the l.h.s. being negative while the r.h.s. is positive. Similarly, the inequality C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≤ y2 − C(x1 , y2 ) , is satisfied for any copula C. A second form of relaxation of the Frank inequality is obtained by setting the two inner copulas in the r.h.s. equal to the smallest copula W , which turns it into the inequality C(x1 , y1 )+C(x2 , y2 )−C(x1 , y2 )−C(x2 , y1 ) ≤ C (min(x2 , 1 − y1 ), min(1 − x1 , y2 )) , which, given a copula C, should be satisfied for all 0 ≤ x1 ≤ x2 ≤ 1 and 0 ≤ y1 ≤ y2 ≤ 1. We now prove that this inequality is satisfied for any radially symmetric copula C. We do so by distinguishing four cases. First, assume that x1 + y2 ≤ 1 and x2 + y1 ≤ 1. Then, the inequality reduces to C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≤ C(x2 , y2 ) , or, equivalently, to C(x1 , y1 ) − C(x1 , y2 ) − C(x1 , y2 ) ≤ 0 , which is clearly satisfied for any copula C. Next, assume that x1 + y2 ≥ 1 and x2 + y1 ≤ 1. Then, the inequality reduces to C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≤ C(x2 , 1 − x1 ) , or, equivalently, to C(x2 , y2 ) − C(x2 , 1 − x1 ) ≤ C(x1 , y2 ) + C(x2 , y1 ) − C(x1 , y1 ) , Now, since any C is 1-Lipschitz continuous and y2 ≥ 1 − x1 , it holds that C(x2 , y2 )−C(x2 , 1−x1 ) ≤ x1 +y2 −1 ≤ C(x1 , y2 ) ≤ C(x1 , y2 )+C(x2 , y1 )−C(x1 , y1 ) , where the last inequality follows from the fact that C(x2 , y1 ) − C(x1 , y1 ) ≥ 0. Hence, the given inequality is satisfied for any copula C. The same conclusion can be drawn in the case where x1 + y2 ≤ 1 and x2 + y1 ≥ 1. Finally, we assume that x1 + y2 ≥ 1 and x2 + y1 ≥ 1, in which case the inequality reduces to C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≤ C(1 − y1 , 1 − x1 ) , The r.h.s. is independent of x2 and y2 , while the l.h.s. is an increasing function of x2 and y2 . Therefore, we only need to verify that the inequality holds for x2 = y2 = 1, i.e., C(x1 , y1 ) ≤ x1 + y1 − 1 + C(1 − x1 , 1 − y1 ) , 16
for all 0 ≤ x1 , y1 ≤ 1. Replacing x1 and y1 respectively by 1 − x1 , 1 − y1 , the converse inequality is obtained, whence the copula C should coincide with its ˆ in other words, C should be radially symmetric. survival copula C, The main theorem of the present paper was concerned with associative copulas only. If we drop the associativity, then we expect more families of copulas to satisfy the Frank equality. Obviously, these copulas still need to be symmetric and radially symmetric, as indicated by Proposition 4.1. We performed numerical experiments for several such families of non-associative copulas, namely: (1) The Farlie-Gumbel-Morgenstern family of copulas, parametrized by θ ∈ [−1, 1]: Cθ (x, y) = xy + θxy(1 − x)(1 − y) . (2) The Fr´echet family of copulas, parametrized by α, β ≥ 0, with α + β ≤ 1: Cα,β = αM + (1 − α − β)Π + βW . (3) The Plackett family of copulas, parametrized by θ ≥ 0: ⎧ 1 ⎪ ⎪ [1 + (θ − 1)(x + y)− ⎪ ⎪ ⎪ 2(θ − 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [1 + (θ − 1)(x + y)]2 − 4xyθ(θ − 1)
Cθ (x, y) = ⎪
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ xy
if θ ≥ 0 , θ = 1 , if θ = 1 .
(4) The family parametrized by θ ∈ [−1, 1] and defined by Cθ (x, y) = xy +
θ [x(1 − x) min(y, 1 − y) + y(1 − y) min(x, 1 − x)] . 2
(5) The family parametrized by θ ∈ [−1, 1] and defined by Cθ (x, y) = xy +
θ [x(1 − x) sin(πy) + y(1 − y) sin(πx)] . 2π
For all of these families, the validity of the Frank inequality has been numerically explored on fine regular grids with intersection points (xi , yj ). In none of these families, however, we were able to identify a copula that violates the Frank inequality. As a by-product of our investigation, we have highlighted the property of super-additivity shared by all Frank copulas. We are not aware whether any systematic study of this property has been done so far. If not, it remains an interesting open problem to characterize the super-additive Archimedean, super-additive associative or super-additive copulas in general; a related open 17
problem raising the question whether super-additive strict t-norms are copulas was also mentioned in [4].
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[32] R. Nelsen, Properties of a one-parameter family of bivariate distributions with specified marginals, Communications in Statistics, Theory and Methods 15 (1986) 3277–3285. [33] R. Nelsen, Copulas and association, In: G. Dall’aglio, S. Kotz, G. Salinetti, Eds., Advances in Probability Distributions with Given Marginals; Beyond the Copulas, Mathematics and its Applications 67, Kluwer Academic Publishers, Dordrecht, pp. 51–74, 1991. [34] R. Nelsen, Dependence and order in families of Archimedean copulas, Journal of Multivariate Analysis 60 (1997) 111–122. [35] R. Nelsen, An Introduction to Copulas. (2nd ed.) Springer, Berlin Heidelberg New York, 2006. [36] B. Van de Walle, B. De Baets and E. Kerre, Characterizable fuzzy preference structures, Annals of Operations Research 80 (1998), 105–136.
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B. De Baets, H. De Meyer
PII: DOI: Reference:
S0165-0114(17)30136-7 http://dx.doi.org/10.1016/j.fss.2017.03.017 FSS 7195
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Fuzzy Sets and Systems
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The Frank inequality B. De Baets a and H. De Meyer b a Department
of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B-9000 Gent, Belgium [email protected]
b Department
of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281 S9, B-9000 Gent, Belgium [email protected]
Abstract We investigate a functional inequality for copulas that has emerged from our study of the comparison of a set of random variables pairwisely coupled by a same copula. Any copula satisfying this inequality is necessarily symmetric and radially symmetric. Moreover, any associative copula satisfying this inequality is a solution to the well-known Frank equation. For this reason, the inequality is coined the Frank inequality. We fully characterize the associative copulas that satisfy the Frank inequality: they turn out to be either Frank copulas or ordinal sums of a same Frank copula with equidistant idempotent elements. As a by-product, we observe that Frank copulas are super-additive on the unit square. Key words: Associative copula, Frank copula, Frank equation, Frank inequality, Super-additivity, Radial symmetry.
1
Introduction
Triangular norms (t-norms for short), originating from the theory of probabilistic metric spaces, have been playing a key role in fuzzy set theory for decades. They have quickly become the unquestioned standard for modelling the pointwise intersection of fuzzy sets as well as for the modelling the conjunction in fuzzy logic [27]. The search for t-norms satisfying additional interesting properties has led to a number of interesting functional equations (see, e.g., [2,3,7,9]), resulting in well-known parametric families of triangular norms and conorms [30]. Also the theory of probabilistic metric spaces has been a source of interesting functional equations. In the past, many investigations were aimed at finding Preprint submitted to Elsevier Science
March 31, 2017
the solution(s) to functional equations in the space of uniform distribution functions [1]. A prominent example is the work of Frank [22,23] on a family of binary operations on the space of probability distribution functions which are related to sums of dependent random variables and are induced by copulas. This led him to the functional equation of simultaneous associativity and to the discovery of a one-parameter family of associative copulas which have turned out to be of great importance in certain areas of statistics [24,25,26,32,33,34]. This equation, however, when restricted to the class of associative copulas, does not uniquely characterize the Frank family of copulas, as it also has as solutions all ordinal sums of Frank copulas. Note that Frank copulas and ordinal sums thereof are more often regarded as tnorms [30] and that in this context the Frank equation has also been studied for the more general classes of uninorms and nullnorms [8]. The Frank t-norm family appears abundantly in fuzzy set theory, a prominent example being fuzzy preference modelling [15,20,21,36]. Recently, we have investigated transitivity properties of certain types of reciprocal relations that are generated by pairwisely comparing (in a probabilistic manner) random variables constituting a random vector. We have studied a variety of couplings between these random variables and our analysis has revealed that the reciprocal relations generated by many of these couplings possess transitivity properties that can be nicely characterized [11,16,17,18,19]. In case the random variables are pairwisely coupled by a same copula [11], the transitivity of the generated reciprocal relation can be captured in the general cycle-transitivity framework [10,13] only if the copula fulfills a countably infinite number of conditions, parametrized by an integer parameter (k ≥ 2), each of them taking the form of a functional inequality. Recently, we have shown that all Frank copulas satisfy each and every of these inequalities [12], and that the fulfillment of the inequality with parameter value k implies that of the inequality with parameter value k , for any 2 ≤ k < k. Realizing in this paper that the fulfillment of the inequality with k = 2 (and hence also with k > 2) trivially implies that of the Frank equation as well [11], inspired us to call this inequality the Frank inequality. In the present paper, we fully characterize the associative copulas that satisfy this inequality. Obviously, they must constitute a particular class of ordinal sums of Frank copulas. The outline of this paper is as follows. In Section 2, we recall what is known about Frank’s functional equation. In Section 3, we prove that every Frank copula is super-additive. Next, in Section 4, we introduce the Frank inequality and recall (while providing an alternative elegant proof) that all Frank copulas fulfill this inequality; moreover, we show that the Frank inequality implies both symmetry and radial symmetry. In Section 5, we fully characterize the class of associative copulas that satisfy the Frank inequality. In the final section, we explore some weaker forms of the Frank inequality and point to some families of non-associative copulas that are conjectured to satisfy the Frank 2
inequality.
2
Copulas and the functional equation of Frank
Copulas have become a basic tool in probability theory and in mathematical statistics. Let X and Y be continuous random variables, defined on a common probability space and taking values in R, with continuous distributions FX and FY and joint distribution HXY , i.e., FX (u) = Prob(X ≤ u) ,
FY (v) = Prob(Y ≤ v) ,
HXY (u, v) = Prob(X ≤ u, Y ≤ v) . Then there exists a unique copula CXY such that HXY (u, v) = CXY (FX (u), FY (v)) . Generally, a function C : [0, 1]2 → [0, 1] is called a copula if it satisfies the boundary conditions C(t, 0) = C(0, t) = 0 and C(t, 1) = C(1, t) = t for all t ∈ [0, 1], and the property of 2-increasingness, i.e., VC ([x1 , y1 ] × [x2 , y2 ]) := C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≥ 0 for all 0 ≤ x1 ≤ x2 ≤ 1 and 0 ≤ y1 ≤ y2 ≤ 1. VC ([x1 , y1 ] × [x2 , y2 ]) is called the C-volume of the rectangle [x1 , y1 ] × [x2 , y2 ]. Note that any copula C is increasing and 1-Lipschitz continuous. Two random variables X and Y are exchangeable if and only if the associated copula C is symmetric, i.e., C(x, y) = C(y, x) for all (x, y) ∈ [0, 1]2 . Any associative copula is symmetric and thus a 1-Lipschitz continuous t-norm. Conversely, any t-norm that is 2-increasing is an associative copula. For more information, we refer to [5]. The dual of a copula C is the function C˜ on [0, 1]2 defined by ˜ y) = x + y − C(x, y) , C(x, whereas the co-copula of C is the function C ∗ on [0, 1]2 defined by C ∗ (x, y) = 1 − C(1 − x, 1 − y) . On the other hand, the function Cˆ on [0, 1]2 defined by ˆ y) = x + y − 1 + C(1 − x, 1 − y) , C(x, 3
is a copula, which is called the survival copula associated to C. Its probabilistic interpretation is given by Prob(X > u, Y > v) = CˆXY (Prob(X > u), Prob(Y > v)) . For continuous random variables X and Y , the identity CXY = CˆXY characterizes the property of radial symmetry (see [35]). Note that CˆXY ≤ CXY implies CXY = CˆXY . In the study of the associativity of certain binary operations on probability distributions [22,23] and in a number of other settings as well, one encounters the functional equation T (x, y) + S(x, y) = x + y
(1)
for any (x, y) ∈ [0, 1]2 , where T is a t-norm and S a t-conorm. A couple (T, S) satisfying (1), with T a t-norm and S a t-conorm, is called a solution to (1). Since S is increasing, T is 1-Lipschitz continuous, whence T must be a copula, ˜ Thus, the problem is to determine all solutions (C, C) ˜ say C, and S must be C. for which C is an associative copula and its dual C˜ is also (simultaneously) associative. To begin, consider the following one-parameter family of strict t-norms (we use the parametrization commonly used in the framework of copulas):
1 (e−αx − 1)(e−αy − 1) Cα (x, y) = − log 1 + , α e−α − 1
α = 0 ,
(2)
and its limits C−∞ = W ,
C0 = Π ,
C+∞ = M ,
with W (x, y) = max(x + y − 1, 0), Π(x, y) = xy and M (x, y) = min(x, y). A straightforward calculation yields x + y − Cα (x, y) = 1 − Cα (1 − x, 1 − y), i.e., C˜α = Cα∗ , and since the associativity of Cα implies that of Cα∗ , it follows ¯ = R∪{−∞, +∞}, the couple (Cα , C ∗ ) is a solution to (1). that, for any α ∈ R α The remarkable and surprising fact is that these couples (Cα , Cα∗ ) and couples ˜ for which C is an ordinal sum of Cα ’s are the only solutions of (1). (C, C) Theorem 2.1 [23] Consider a t-norm T and a t-conorm S. Then the couple (T, S) is a solution to (1) if and only if T is an ordinal sum of Frank t-norms and S = T˜. Note that if (T, S) is a solution to (1) for a proper ordinal sum T , then S = T ∗ if and only if the set of idempotent elements of T is symmetric with respect to the point x = 1/2. In the language of copulas, the latter is usually presented as follows. 4
Theorem 2.2 [31] An associative copula C is radially symmetric (i.e., it ˆ if and only if it is a Frank copula or a symmetric ordinal holds that C = C) sum of Frank copulas, i.e., an ordinal sum of the form C = ( ak , bk , Cαk )k∈K , where for each k ∈ K, there exists a k ∈ K such that αk = αk and ak + bk = bk + ak = 1.
3
Super-additivity of Frank copulas
The family of Frank t-norms has been widely studied, especially in the fuzzy set community. Regarded as a family of copulas, the Frank copulas have been extensively discussed in statistics [24,28,29,32]. They have many interesting properties, among which the property of super-additivity that we will use in the proof of our main theorem. We are not aware that this property has been emphasized before in the literature. We present it here as a lemma. Lemma 3.1 Any Frank copula Cα is super-additive on [0, 1]2 , i.e., for any x1 , x2 , y1 , y2 ∈ [0, 1] such that x1 + x2 ≤ 1 and y1 + y2 ≤ 1, it holds that Cα (x1 + x2 , y1 + y2 ) ≥ Cα (x1 , y1 ) + Cα (x2 , y2 ) .
(3)
¯ For i = 1, 2, we introduce Proof Consider a Frank copula Cα , with α ∈ R. the notations e−αxi − 1 e−αyi − 1 , vi = −α , ui = −α e −1 e −1 and t = e−α − 1 . Then, ui , vi ∈ [0, 1], while t > 0 if α < 0 and −1 < t < 0 if α > 0. Furthermore, t → −1 if α → +∞, t → +∞ if α → −∞, and t → 0 if α → 0. For i = 2, Cα (xi , yi ) can then be expressed as Cα (xi , yi ) = −
1 ln(1 + tui vi ) , α
while Cα (x1 + x2 , y1 + y2 ) = −
1 ln[1 + t(u1 + u2 + tu1 u2 )(v1 + v2 + tv1 v2 )] . α
Substituting these expressions in (3), it follows that we must prove that the inequality t2 u1 u2 v1 v2 + t[u1 u2 (v1 + v2 ) + v1 v2 (u1 + u2 ) − u1 u2 v1 v2 ] + u1 v2 + u2 v1 ≥ 0 (4) holds for any u1 , u2 , v1 , v2 ∈ [0, 1] and any t ≥ −1. The l.h.s. of the latter inequality is linear in u1 . For u1 = 0, it simplifies to u2 v1 (1+v2 t), an expression 5
that is non-negative for any u2 , v1 , v2 ∈ [0, 1] and any t ≥ −1. For u1 = 1, it reduces to the expression t2 u2 v1 v2 + t[u2 (v1 + v2 ) + v1 v2 (1 + u2 ) − u2 v1 v2 ] + v2 + u2 v1 ,
(5)
which can be regarded as a linear function of u2 . For u2 = 0, the latter expression simplifies to v2 (1 + v1 t), an expression that is non-negative for any v1 , v2 ∈ [0, 1] and any t ≥ −1. For u2 = 1, it reduces to the expression (t+1)(v1 +v2 +v1 v2 t), again an expression that is non-negative for any v1 , v2 ∈ [0, 1] and any t ≥ −1. Hence, (5) is non-negative for any u2 , v1 , v2 ∈ [0, 1] and any t ≥ −1, which proves that inequality (4) holds for all u1 , u2 , v1 , v2 ∈ [0, 1] and all t ≥ −1. Since the Frank copulas are radially symmetric, the property of super-additivity can be reshaped in the following form. Lemma 3.2 For any Frank copula Cα , the inequality Cα (x1 + x2 − 1, y1 + y2 − 1) ≥ Cα (x1 , y1 ) + Cα (x2 , y2 ) − 1
(6)
holds for any x1 , x2 , y1 , y2 ∈ [0, 1] such that x1 + x2 ≥ 1 and y1 + y2 ≥ 1. ¯ Its radial symmetry is Proof Consider a Frank copula Cα , with α ∈ R. expressed by the equality Cα (x, y) = Cα (1 − x, 1 − y) + x + y − 1 , which holds for any (x, y) ∈ [0, 1]2 . When applied to inequality (3), the latter can be brought into the equivalent form Cα (1 − x1 − x2 , 1 − y1 − y2 ) + x1 + x2 + y1 + y2 − 1 ≥ Cα (1 − x1 , 1 − y1 ) + Cα (1 − x2 , 1 − y2 ) + x1 + y1 + x2 + y2 − 2 , or Cα (1 − x1 − x2 , 1 − y1 − y2 ) ≥ Cα (1 − x1 , 1 − y1 ) + Cα (1 − x2 , 1 − y2 ) − 1 . If we replace 1 − xi by xi and 1 − yi by yi (i = 1, 2), then x1 + x2 ≥ 1 and y1 + y2 ≥ 1, which proves that under the conditions stated, inequality (6) is indeed satisfied.
4
The Frank inequality
An intuitively appealing way of comparing two random variables is to consider the probability that the first one takes a value greater than or equal to 6
the second one. More formally, given a random vector (X1 , X2 , . . . , Xn ), we consider the [0, 1]-valued relation Q defined by Q(Xi , Xj ) = Prob{Xi > Xj } +
1 Prob{Xi = Xj } . 2
This relation is reciprocal, i.e., Q(Xi , Xj ) + Q(Xj , Xi ) = 1 for all i, j ∈ {1, . . . , n}. In [16,17,18,19], we studied the transitivity of this reciprocal relation in the special case that the coupling between any two components of the random vector is modelled by a same copula C, where C is either W , Π or M . In [11], we further pointed out that for a general copula C the transitivity of the reciprocal relation Q can only be captured in the cycle-transitivity framework [13,14] provided the copula C satisfies a countably infinite number of inequalities. More explicitly, we require C to satisfy the following family of inequalities: i
C(xi , yi ) − ⎛
≤ C ⎝yk +
xk +
j
j
C(xk−2j−1 , yk−2j−2 ) −
j
C(xk−2j , yk−2j−1 ) −
j
C(xk−2j−2 , yk−2j−1 ) −
j
C(xk−2j−1 , yk−2j ),
j
C(xk−2j−1 , yk−2j )
⎞
C(xk−2j , yk−2j−1 )⎠
(7)
for an arbitrary integer k > 1 and for all 0 ≤ x1 ≤ x2 ≤ · · · ≤ xk ≤ 1 and 0 ≤ y1 ≤ y2 ≤ · · · ≤ yk ≤ 1, and where the sums extend over all integer values that lead to meaningful indices of x and y. We refer to the first (i.e., k = 2) and simplest inequality of this family as the Frank inequality because, as we will prove later on, the only associative copulas that satisfy this inequality are the Frank copulas and very specific ordinal sums of Frank copulas. Note that if a copula C satisfies the Frank inequality, this means that the C-volume of any rectangle [x1 , x2 ] × [y1 , y2 ] in the unit square is bounded from above by the C-volume of the rectangle [0, x ] × [0, y ], where x = y2 − C(x1 , y2 ) and y = x2 − C(x2 , y1 ). Definition 4.1 A copula C : [0, 1]2 → [0, 1] satisfies the Frank inequality if for any 0 ≤ x1 ≤ x2 ≤ 1 and any 0 ≤ y1 ≤ y2 ≤ 1, it holds that C(x1 , y1 )+C(x2 , y2 )−C(x1 , y2 )−C(x2 , y1 ) ≤ C(y2 −C(x1 , y2 ), x2 −C(x2 , y1 )) . (8) The following proposition shows that copulas satisfying the Frank inequality have some interesting properties. 7
Proposition 4.1 Any copula C that satisfies the Frank inequality has the following properties: (i) C is symmetric; (ii) C is radially symmetric. Proof To prove (i), we set x1 = 0 and y1 = 0 in (8). The inequality then reduces to C(x2 , y2 ) ≤ C(y2 , x2 ), from which the symmetry of C immediately follows. To prove (ii), we set x2 = y2 = 1 in (8). Using the symmetry of C, it then follows that C(x1 , y1 ) + 1 − x1 − y1 ≤ C(1 − x1 , 1 − y1 ). Replacing herein x1 by 1 − x1 and y1 by 1 − y1 , it follows that the inequality turns into an equality. Hence, C is radially symmetric. We have already proven in [11] that all Frank copulas satisfy the Frank inequality. For the sake of completeness, we provide here an alternative and short proof. Proposition 4.2 [11] The Frank copulas satisfy the Frank inequality. ¯ We use the notations introProof Consider a Frank copula Cα , with α ∈ R. duced in the proof of Lemma 3.1. Since Cα (xi , yj ) = −
1 log [1 + tui vj ] , α
e−α(x2 −Cα (x2 ,y1 )) = (1 + tu2 )/(1 + tu2 v1 ) , and e−α(y2 −Cα (x1 ,y2 )) = (1 + tv2 )/(1 + tu1 v2 ) , inequality (8) is equivalent to t(1 + tu1 v1 )(1 + tu2 v2 ) ≤ t(1 + tu2 v1 )(1 + tu1 v2 ) + t2 (1 − u1 )(1 − v1 )u2 v2 . This inequality is trivially satisfied if u2 = 0 or v2 = 0. Assume that u2 = 0 and v2 = 0, then the inequality is equivalent to
t2 u2 v2 (1 − u1 )(1 − v1 ) − 1 −
u1 u2
1−
v1 v2
≥ 0.
Since the latter inequality clearly holds for any 0 ≤ u1 ≤ u2 ≤ 1 any 0 ≤ v1 ≤ ¯ v2 ≤ 1 and any t ≥ −1, Cα satisfies (8) for any α ∈ R. 8
5
Characterization of the associative copulas that satisfy the Frank inequality
In this section, similar to the original Frank equation, we restrict our attention to associative operations. When restricting to associative copulas, Theorem 2.2, Proposition 4.1 and Proposition 4.2 already indicate that the solution set consists of the Frank copulas, possibly complemented with certain symmetric ordinal sums of Frank copulas. The following theorem, the main result of this paper, identifies the structure of these ordinal sum solutions, and thus lays bare the full solution set. Theorem 5.1 An associative copula C satisfies the Frank inequality if and only if it is a Frank copula or an ordinal sum of the form C = ( i/k, (i + 1)/k, Cα )i∈I where I = {0, 1, . . . , k − 1}, for some integer k > 1, and Cα is a Frank copula. Proof As it is already known that Frank copulas are solutions, we can restrict our attention to symmetric ordinal sums of Frank copulas. Part 1. Suppose that C is such an ordinal sum that satisfies the Frank inequality. We first prove that the idempotent elements of C are equally spaced. If there is a single idempotent element u, then it must equal 12 , since 1−u is an idempotent element as well. If there are two idempotent elements u < v, then C(u, u) = C(u, v) = u and C(v, v) = v. Setting x1 = y1 = u and x2 = y2 = v in (8) yields the condition v − u ≤ C(v − u, v − u), which proves that v − u is also an idempotent element of C. In particular, if u < 1/2 and choosing v = 1 − u, it follows that also 1 − 2u and 2u are idempotent elements of C. Proceeding in the same way, we obtain that if u is an idempotent element of C, then all elements of the form ku, with k ∈ {1, 2, . . . , 1/u }, are idempotent elements as well. If C has an idempotent element other than 0 and 1 and the number of idempotent elements is finite, then there exists a smallest idempotent element u = 0, whence 1 − u is then necessarily the largest idempotent element different from 1. It thus follows that there must exist an integer n ≥ 1 such that nu = 1 − u, hence u = 1/(n + 1), a fractional number. Consequently, the set of idempotent elements of C is given by {i/(n + 1) | i ∈ {0, 1, . . . , n + 1}}. If C has an infinite number of idempotent elements, then the compactness of [0, 1] implies that there is at least one element in [0, 1] that is an accumulation point of idempotent elements (note that this accumulation point is not necessarily an idempotent element itself). But, since differences of idempotent elements yield again idempotent elements, 0 must be an accumulation 9
point. As all multiples of idempotent elements yield idempotent elements, any x ∈ [0, 1] is an accumulation point of idempotent elements of C. Hence, if the set of idempotent elements of C is countably infinite, then this set is dense in [0, 1]. Thus, the diagonal section of C coincides with the identity function on a dense subset of [0, 1], and due to its continuity, it coincides with the identity function on [0, 1]. Since M is the only copula with the identity function as diagonal section, it holds that C = M and thus not a proper ordinal sum. We can thus safely restrict our attention to ordinal sums C of the form C = ( i/k, (i + 1)/k, Cαi )i∈I , where I = {0, 1, . . . , k − 1}. We prove that all copulas Cαi must be equal to a same Frank copula. Let us set x1 = y1 = i/k in (8) for an arbitrary but fixed i ∈ {0, 1, . . . , k − 1}, and let x2 , y2 ∈ [i/k, (i + 1)/k]. The inequality then reduces to
1 i i Cαi (kx2 − i, ky2 − i) ≤ C x2 − , y2 − k k k
=
1 Cα (kx2 − i, ky2 − i) . k 0
Hence, for all i ∈ {1, 2, . . . , k − 1}, it holds that C α i ≤ C α0 . Let us further set x2 = y2 = (i + 1)/k in (8) for arbitrary but fixed i ∈ {0, . . . , k − 1}, and let x1 , y1 ∈ [0, 1/k]. The inequality now reduces to
i+1 1 i+1 i+1 Cα (kx1 , ky1 ) + − x1 − y1 ≤ C − x1 , − y1 k 0 k k k =
1 i + Cαi (1 − kx1 , 1 − ky1 ) . k k
(9)
In particular, we obtain for i = 0 that Cα0 (kx1 , ky1 ) + 1 − kx1 − ky1 ≤ Cα0 (1 − kx1 , 1 − ky1 ) , from which it follows that Cα0 is a radially symmetric associative copula. Since Cα0 cannot have idempotent elements other than 0 and 1, Cα0 must be a Frank copula different from M , i.e., α0 = +∞). Due to the radial symmetry of Cα0 , inequality (9) can be rewritten as Cα0 (1 − kx1 , 1 − ky1 ) ≤ Cαi (1 − kx1 , 1 − ky1 ) . Hence, it must hold for all i ∈ {1, 2, . . . , k − 1} that Cα 0 ≤ C α i . 10
It follows that Cαi = Cα for all i ∈ {0, 1, . . . , k − 1}. Hence, we have shown that the ordinal sum solutions are of the type mentioned in the theorem. Part 2. Conversely, we prove that any ordinal sum of the mentioned type satisfies the Frank inequality. Consider a copula C of the form C = ( i/k, (i + 1)/k, Cα )i∈I where I = {0, 1, . . . , k − 1}, for some integer k > 1, and Cα is a Frank copula. The part of the unit square where the copula C coincides with a rescaled Frank copula Cα consists of k squares of size 1/k aligned along the diagonal. Outside these squares the copula C coincides with M . The l.h.s. of the Frank inequality represents the C-volume of an arbitrary rectangle [x1 , x2 ] × [y1 , y2 ] with corner points (x1 , y1 ), (x2 , y2 ), (x1 , y2 ) and (x2 , y1 ). We distinguish the following situations covering all possibilities: (1) The four corner points fall inside a same square. (2) Two neighbouring corner points fall in a same square, the two other corner points are outside the squares. (3) Two opposite corner points fall in different squares, the other two (opposite) corner points are outside the squares. (4) One corner point falls inside a square, the three other corner points are outside the squares. (5) None of the corner points falls inside a square. Situation 1. Let us assume that i i+1 ≤ x1 ≤ x2 ≤ , k k
i i+1 ≤ y 1 ≤ y2 ≤ , k k
for arbitrary but fixed i ∈ {0, 1, . . . , k − 1}. We must verify whether the Frank inequality 1 1 Cα (kx1 − i, ky1 − i) + Cα (kx2 − i, ky2 − i) k k 1 1 − Cα (kx1 − i, ky2 − i) − Cα (kx2 − i, ky1 − i) k k 1 1 i i ≤ C(y2 − − Cα (kx1 − i, ky2 − i), x2 − − Cα (kx2 − i, ky1 − i)) k k k k is satisfied. Both arguments of the copula C in the r.h.s. can take any value in the interval [0, 1/k]. Hence, C should be identified with the copula Cα rescaled to the first square, i.e., 11
C(y2 − =
i i 1 1 − Cα (kx1 − i, ky2 − i), x2 − − Cα (kx2 − i, ky1 − i)) k k k k
1 Cα (ky2 − i − Cα (kx1 − i, ky2 − i), kx2 − i − Cα (kx2 − i, ky1 − i)) . k
Then the above inequality reduces to the Frank inequality for the copula Cα . This inequality is indeed satisfied for Frank copulas as is stated in Proposition 4.2. Situation 2. Let us first assume that the right corner points (x2 , y1 ) and (x2 , y2 ) of the given rectangle fall inside a same square, i.e., x1 <
i i+1 ≤ x2 ≤ , k k
i+1 i ≤ y1 ≤ y2 ≤ , k k
for arbitrary but fixed i ∈ {1, 2, . . . , k − 1}. The C-volume of the rectangle [x1 , x2 ] × [y1 , y2 ] in the l.h.s. of the Frank inequality equals the C-volume of the smaller rectangle [i/k, x2 ] × [y1 , y2 ] with all its corner points situated in the same square [i/k, (i + 1)/k], [i/k, (i + 1)/k]. Making use of the result of Situation 1, the Frank inequality holds for the latter C-volume. It follows that
C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) = C(i/k, y1 ) + C(x2 , y2 ) − C(i/k, y2 ) − C(x2 , y1 ) ≤ C(x2 − C(x2 , y1 ), y2 − C(i/k, y2 )) ≤ C(x2 − C(x2 , y1 ), y2 − C(x1 , y2 )) , where the last inequality follows from the fact that x1 ≤ i/k. This proves that for this specific case, the Frank inequality is satisfied. Due to the symmetry of C, the same can be immediately concluded for rectangles with the lower corner points falling inside a same square. Let us next assume that the left corner points (x1 , y1 ) and (x1 , y2 ) fall inside a same square, i.e., i i+1 ≤ x1 ≤ ≤ x2 , k k
i i+1 ≤ y 1 ≤ y2 ≤ , k k
for arbitrary but fixed i ∈ {1, 2, . . . , k − 1}. We now have 12
C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) = C(x1 , y1 ) + C((i + 1)/k, y2 ) − C(x1 , y2 ) − C((i + 1)/k, y1 ) ≤ C((i + 1)/k − C((i + 1)/k, y1 ), y2 − C(x1 , y2 )) ≤ C(x2 − C(x2 , y1 ), y2 − C(x1 , y2 )) , where the last inequality follows from the fact that for any copula C, the function x − C(x, y) is a copula, and is thus increasing in both arguments x and y. This concludes the proof that the Frank inequality is generally satisfied for Situation 2. Situation 3. This is the situation where the opposite corner points (x1 , y1 ) and (x2 , y2 ) fall inside different squares. Clearly, the other two corner points then fall outside any square. Hence, let us assume that i i+1 ≤ x 1 , y1 ≤ , k k
j+1 j ≤ x2 , y2 ≤ , k k
for arbitrary but fixed i, j ∈ {1, 2, . . . , k − 1} and 0 ≤ i < j ≤ k − 1. It follows that j−i−1 j−i+1 ≤ x2 − y 1 ≤ , k k
j−i+1 j−i−1 ≤ y2 − x 1 ≤ . k k
The Frank inequality reduces to j i 1 1 + Cα (kx1 − i, ky1 − i) + + Cα (kx2 − j, ky2 − j) − x1 − y1 k k k k ≤ C(x2 − y1 , y2 − x1 ) . or, equivalently, to Cα (kx1 −i, ky1 −i)+Cα (kx2 −j, ky2 −j) ≤ kx1 +ky1 −i−j+kC(x2 −y1 , y2 −x1 ) . We distinguish the following subcases. j−i−1 j−i−1 j−i j−i ≤ x2 − y 1 ≤ and ≤ y 2 − x1 ≤ . k k k k In this case, the Frank inequality becomes
(a) The case
Cα (kx1 − i, ky1 − i) + Cα (kx2 − j, ky2 − j) ≤ kx1 + ky1 − 2i − 1 +Cα (kx2 − j − (ky1 − i) + 1, ky2 − j − (kx1 − i) + 1) . 13
Introducing the notations s1 = kx1 − i ,
s2 = kx2 − j ,
t1 = ky1 − i ,
t2 = ky2 − j ,
the above is equivalent to the inequality Cα (s1 , t1 ) + Cα (s2 , t2 ) ≤ s1 + t1 − 1 + Cα (s2 − t1 + 1, t2 − s1 + 1) , or, since Cα is radially symmetric, the above is also equivalent to Cα (s2 , t2 ) + Cα (1 − s1 , 1 − t1 ) ≤ Cα (s2 − t1 + 1, t2 − s1 + 1) , which should be satisfied for all 0 ≤ s2 ≤ t1 ≤ 1 and all 0 ≤ t2 ≤ s1 ≤ 1. Due to the super-additivity property of the Frank copula Cα (see Lemma 3.1), this inequality is indeed satisfied. j−i j−i j−i+1 j−i+1 ≤ x 2 − y1 ≤ and ≤ y2 − x1 ≤ . k k k k Now, the Frank inequality reduces to
(b) The case
Cα (kx1 − i, ky1 − i) + Cα (kx2 − j, ky2 − j) ≤ kx1 − i + ky1 − i + Cα (kx2 − j − (ky1 − i), ky2 − j − (kx1 − i)) . Introducing the notations s1 = kx1 − i ,
s2 = kx2 − j ,
t1 = ky1 − i ,
t2 = ky2 − j ,
the above is equivalent to the inequality Cα (s1 , t1 ) + Cα (s2 , t2 ) ≤ s1 + t1 + Cα (s2 − t1 , t2 − s1 ) , which should be fulfilled for all 0 ≤ t1 ≤ s2 ≤ 1 and for all 0 ≤ s1 ≤ t2 ≤ 1. Making use of the radial symmetry of Cα , this inequality can be rewritten as Cα (s2 , t2 ) + Cα (1 − t1 , 1 − s1 ) ≤ 1 + Cα (s2 + (1 − t1 ) − 1, t2 + (1 − s1 ) − 1) , which should be satisfied for any s2 + 1 − t1 ≥ 1 and any t2 + 1 − s1 ≥ 1. Again, due to the super-additivity property of the Frank copula Cα (see Lemma 3.1), this inequality is indeed satisfied. j−i−1 j−i j−i+1 j−i ≤ x2 − y 1 ≤ and ≤ y 2 − x1 ≤ k k k k Now, the Frank inequality reduces to
(c) The case
Cα (kx1 − i, ky1 − i) + Cα (kx2 − j, ky2 − j) ≤ kx1 − i + kx2 − j , which is trivially fulfilled for any copula.
14
j−i j−i+1 j−i j−i−1 ≤ x 2 − y1 ≤ and ≤ y 2 − x1 ≤ k k k k As before, in this case the Frank inequality is trivially satisfied.
(d) The case
Situation 4. Repeating the arguments by which Situation 2 was essentially reduced to Situation 1, the present situation can be similarly reduced either to Situation 1 or Situation 3. The proof is therefore immediate. Situation 5. Depending on the location of the corner points, we can rely either on the results obtained for Situation 1 or Situation 3, or we can make use of the fact that the C-volume of the given rectangle is zero (that is, when the rectangle does not enclose any of the squares centered along the diagonal), in which case the Frank inequality is trivially satisfied. Note that all Frank copulas are Archimedean copulas, but ordinal sums of Frank copulas are not. Therefore, Theorem 5.1 yields the following corollary. Corollary 5.1 The Frank copulas are the only Archimedean copulas that satisfy the Frank inequality.
6
Discussion
Leaving aside the solutions that are proper ordinal sums, the only associative copulas that satisfy the Frank inequality are the Frank copulas. As the Frank inequality therefore appears to be tight, it is worthwhile to investigate weaker forms of it. One such form is obtained by setting the outer copula in the r.h.s. equal to the greatest copula M . The inequality then becomes C(x1 , y1 )+C(x2 , y2 )−C(x1 , y2 )−C(x2 , y1 ) ≤ min (x2 − C(x2 , y1 ), y2 − C(x1 , y2 )) , which, given a copula C, should be satisfied for all 0 ≤ x1 ≤ x2 ≤ 1 and 0 ≤ y1 ≤ y2 ≤ 1. However, it is easily verified that any copula C satisfies this inequality. Indeed, the inequality C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≤ x2 − C(x2 , y1 ) , is equivalent to C(x1 , y1 ) − C(x1 , y2 ) ≤ x2 − C(x2 , y2 ) , 15
which is satisfied for any copula C, the l.h.s. being negative while the r.h.s. is positive. Similarly, the inequality C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≤ y2 − C(x1 , y2 ) , is satisfied for any copula C. A second form of relaxation of the Frank inequality is obtained by setting the two inner copulas in the r.h.s. equal to the smallest copula W , which turns it into the inequality C(x1 , y1 )+C(x2 , y2 )−C(x1 , y2 )−C(x2 , y1 ) ≤ C (min(x2 , 1 − y1 ), min(1 − x1 , y2 )) , which, given a copula C, should be satisfied for all 0 ≤ x1 ≤ x2 ≤ 1 and 0 ≤ y1 ≤ y2 ≤ 1. We now prove that this inequality is satisfied for any radially symmetric copula C. We do so by distinguishing four cases. First, assume that x1 + y2 ≤ 1 and x2 + y1 ≤ 1. Then, the inequality reduces to C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≤ C(x2 , y2 ) , or, equivalently, to C(x1 , y1 ) − C(x1 , y2 ) − C(x1 , y2 ) ≤ 0 , which is clearly satisfied for any copula C. Next, assume that x1 + y2 ≥ 1 and x2 + y1 ≤ 1. Then, the inequality reduces to C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≤ C(x2 , 1 − x1 ) , or, equivalently, to C(x2 , y2 ) − C(x2 , 1 − x1 ) ≤ C(x1 , y2 ) + C(x2 , y1 ) − C(x1 , y1 ) , Now, since any C is 1-Lipschitz continuous and y2 ≥ 1 − x1 , it holds that C(x2 , y2 )−C(x2 , 1−x1 ) ≤ x1 +y2 −1 ≤ C(x1 , y2 ) ≤ C(x1 , y2 )+C(x2 , y1 )−C(x1 , y1 ) , where the last inequality follows from the fact that C(x2 , y1 ) − C(x1 , y1 ) ≥ 0. Hence, the given inequality is satisfied for any copula C. The same conclusion can be drawn in the case where x1 + y2 ≤ 1 and x2 + y1 ≥ 1. Finally, we assume that x1 + y2 ≥ 1 and x2 + y1 ≥ 1, in which case the inequality reduces to C(x1 , y1 ) + C(x2 , y2 ) − C(x1 , y2 ) − C(x2 , y1 ) ≤ C(1 − y1 , 1 − x1 ) , The r.h.s. is independent of x2 and y2 , while the l.h.s. is an increasing function of x2 and y2 . Therefore, we only need to verify that the inequality holds for x2 = y2 = 1, i.e., C(x1 , y1 ) ≤ x1 + y1 − 1 + C(1 − x1 , 1 − y1 ) , 16
for all 0 ≤ x1 , y1 ≤ 1. Replacing x1 and y1 respectively by 1 − x1 , 1 − y1 , the converse inequality is obtained, whence the copula C should coincide with its ˆ in other words, C should be radially symmetric. survival copula C, The main theorem of the present paper was concerned with associative copulas only. If we drop the associativity, then we expect more families of copulas to satisfy the Frank equality. Obviously, these copulas still need to be symmetric and radially symmetric, as indicated by Proposition 4.1. We performed numerical experiments for several such families of non-associative copulas, namely: (1) The Farlie-Gumbel-Morgenstern family of copulas, parametrized by θ ∈ [−1, 1]: Cθ (x, y) = xy + θxy(1 − x)(1 − y) . (2) The Fr´echet family of copulas, parametrized by α, β ≥ 0, with α + β ≤ 1: Cα,β = αM + (1 − α − β)Π + βW . (3) The Plackett family of copulas, parametrized by θ ≥ 0: ⎧ 1 ⎪ ⎪ [1 + (θ − 1)(x + y)− ⎪ ⎪ ⎪ 2(θ − 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ [1 + (θ − 1)(x + y)]2 − 4xyθ(θ − 1)
Cθ (x, y) = ⎪
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ xy
if θ ≥ 0 , θ = 1 , if θ = 1 .
(4) The family parametrized by θ ∈ [−1, 1] and defined by Cθ (x, y) = xy +
θ [x(1 − x) min(y, 1 − y) + y(1 − y) min(x, 1 − x)] . 2
(5) The family parametrized by θ ∈ [−1, 1] and defined by Cθ (x, y) = xy +
θ [x(1 − x) sin(πy) + y(1 − y) sin(πx)] . 2π
For all of these families, the validity of the Frank inequality has been numerically explored on fine regular grids with intersection points (xi , yj ). In none of these families, however, we were able to identify a copula that violates the Frank inequality. As a by-product of our investigation, we have highlighted the property of super-additivity shared by all Frank copulas. We are not aware whether any systematic study of this property has been done so far. If not, it remains an interesting open problem to characterize the super-additive Archimedean, super-additive associative or super-additive copulas in general; a related open 17
problem raising the question whether super-additive strict t-norms are copulas was also mentioned in [4].
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