Bivariate copulas, norms and non-exchangeability

Abstract The present paper is related to the study of asymmetry for copulas by introducing functionals based on different norms for continuous variables. In particular, we discuss some facts concerning asymmetry and we point out some flaws occurring in the recent literature dealing with this matter.


Introduction
Bivariate copulas (see for example Nelsen [11] for basic de nitions and results) have been studied in details since more than half century. More recently (see Nelsen [12] and Klement and Mesiar [10]) a rst measure of asymmetry has been considered; several recent papers deal with the largest asymmetry (in this sense) that some relevant families of copulas can reach (see for example Alvoni et al. [2]). For a general reference on asymmetry measures we refer for example to Genest and Neslehová [9]. Concerning other recent papers on the subject, see for example [1,3,6,8]. For a recent reference concerning copulas in general, we indicate the book by Durante and Sempi [7].
A possible set of natural axioms that a measure of asymmetry should satisfy has been indicated in Durante et al. [5]; in particular, a few measures of this type can be obtained by using some classical norms.
In Siburg and Stoimenov [14], an attempt to obtain in a new way, by any norm, a measure of asymmetry (not only for copulas, and not necessarily satisfying all the axioms indicated in Durante et al. [5]), has been done. Unfortunately, several facts indicated there are awed.
Here we discuss the results in Siburg and Stoimenov [14]; we also indicate which additional properties of the norm imply some "good" properties for the corresponding measure of asymmetry de ned there.
The plan of this paper is the following. In the next Section 2 we recall the de nitions concerning copulas and asymmetry. In Section 3, we consider the functional connected with asymmetry, based on norms, introduced in Siburg and Stoimenov [14]: we indicate, by several examples, some mistakes contained there. In Section 4 we discuss some variations concerning the same functional, by using norms satisfying a few additional, simple properties. Finally, Section 5 contains a short discussion concerning the use of norms in the context of copulas.

De nitions and Notations
Set I = [ , ], so I = [ , ] × [ , ]. For the sake of simplicity, from now on, V will denote the vector space of all continuous real functions on I . For f ∈ V, set A (bivariate) copula is an element of V satisfying: In particular, condition (2.5), usually called − increasingness, together with (2.3) implies: A copula can be seen as the restriction to the unit square of a probability distribution function with uniform marginals on [ , ]. We shall denote by C the set of all copulas: this is a convex subset of V.
If C ∈ C, then also C T and C are in C (they are called the transpose, and -respectively-the survival copula associated with C). A copula is said to be symmetric if C = C T .
As known, the following (symmetric) copulas play an important role: W(x, y) = max{x + y − , } ; (2.7) Π(x, y) = xy ; (2.8) Recall that for any copula C we have According to Durante et al. [5], a measure of non-exchangeability for C is a function µ: As examples of such measures, the following ones were indicated in Durante et al. [5]: By the way, µ∞ is exactly the measure of asymmetry used in Nelsen [12] and in Klement and Mesiar [10]. Also, if we denote by ||.||p the classical Lp norms (in I ), we have: Note that all p-norms ( ≤ p ≤ ∞), applied to all functions f on V, satisfy the following properties: Moreover, for every copula C:
For every f ∈ V \ {0} (0 denoting the null element of V, i.e. the identically null function) set Concerning δ, the following properties were indicated (see Siburg and Stoimenov [14, Theorem 2.4]): These properties are not completely natural. Also, we could extend the de nition by setting δ(0) = , so avoiding exclusions in a few cases.
While (P5) is trivially true, the other properties are doubtful, also when f is a copula, unless we require that the norm satis es some additional properties. We discuss this by means of examples; in the rst one we consider functions which are not copulas. The second one has been partly inspired by an example, called "exotic", in Durante et al. [5].
In the previous example, we dealt with functions in V which are not copulas.
The following examples instead are based on copulas: they show that (also for them) (P3), as well (P1)  x, y ∈ I}. A copula K with these properties has been considered for example in Nelsen [11], where the last equality (||K − K T ||∞ = / ) was shown.
Let δa(f ) be the corresponding functional, de ned on V \ {0} according to (3.1). Now consider the copula K de ned before this example. We have: Thus

So, for a suitable value of β, we can have δ β (K) = − (but certainly K ≠ −K T ): thus (P3) (as well as (P1)) is violated.
The same happens if in the denominator we put, instead of ||K|| β , for example:

Some positive results
For any f ≠ 0, the property ||f || = ||f T || is equivalent to δ(f ) = δ(f T ) (compare with Theorem 2.6 (i) in Siburg and Stoimenov [14]). Assume that a norm satis es (N1). Then we obtain for the corresponding δ (see (3.1)): But under the same assumption, we can also prove the following.
If the norm does not satisfy ||f || = ||f T ||, then we can have (P1), for example, if we de ne (instead of using (3.1)): or we change in some other suitable way the denominator. It is simple to see that With such modi cation (P2) and (P3) hold. In fact (concerning δ in (4.3)) we have Similar facts hold if we change the denominator in the other two ways 4 max {||f || , ||f T || }, or (||f ||+ ||f T ||) .
Therefore ||K + K T ||γ = ; ||K − K T ||γ = / + ( / )γ, and then The same considerations hold if at the denominator we put for example This shows that, if we consider all functionals δ derived by norms satisfying (N1), the union of ranges already for the set of copulas is (− , ].

Remark 3.
We observe the following fact. Let a norm satisfy (N2); note that, for any copula C, we have pointwise: |C − C T | ≤ max{C, C T } ≤ C + C T ; then we have ≤ δ(C) for all copulas. This estimate is sharp (see Remark 2). Remark 4. Note that when (P1) holds, we can also "renormalize" δ so that its range become, for example, [0, 1]: however the value would not correspond in general to f symmetric.
Concerning copulas, a better way to "renormalize" would be the following. Assume that a norm satis es  Note that C − C T c = / ≠ || C − C T ||c = / .

nal discussion and concluding remarks
Concerning copulas, the use of norms seems to be not a straight way to deal with; a discussion was done in Darsow and Olsen [4], where also some problems were indicated. Later on, very few papers considered this matter: see for example Siburg and Stoimenov [14]. We recall that, in particular, the use of the Sobolev scalar product for copulas was considered by Siburg and Stoimenov [13]. This is due to more reasons, we indicate some of them. First, copulas do not form a subspace of V (nor a vector space), and the subspace they generate has a non evident structure (see Darsow and Olsen [4,Section 2]). Second, we usually think of properties that the most used norms for the space of continuous functions have, but can be missed; for this reason we are led to think and hope for properties that only additional requirements on the norms imply. So, due to this, not all the norms are appropriate to obtain "nice" measures of asymmetry, also when limited to copulas. In particular, we have shown that in general the functional discussed in Section 3 is not so suitable to measure asymmetry of copulas: this attempt to use norms and extend the study of asymmetry to a larger class of functions needs special care. Start from a norm and de ne symmetry, also in V, by using the simple formula (2.13); clearly (A2), (A3), (A5) (and (P5)) are satis ed (but not necessarily (A4): see Example 4.3). As we can understand, this is not true in general for δ: C and C T play an asymmetric role in its de nition. We recall that the de nition of δ was indicated by Genest and Neslehová in [9, p.94], with the following sentence: "this measure, for the uniform norm, is equivalent to µ∞, so is not considered further here". Indeed, in that case δ = − µ ∞ , but this equivalence is limited to a very particular situation. For example: when C = C T (so µ(C) = ), δ(C) can be both positive and negative (see Example 4.2).