Maximal asymmetry of bivariate copulas and consequences to measures of dependence

: In this article, we focus on copulas underlying maximal non - exchangeable pairs ( ) X Y , of con tinuous random variables X Y , either in the sense of the uniform metric ∞ d or the conditioning - based metrics D p , and analyze their possible extent of dependence quanti ﬁ ed by the recently introduced depen dence measures ζ 1 and ξ . Considering maximal ∞ d - asymmetry we obtain ⎡⎣ ⎤⎦ ∈ ζ , 1 1 5 6 and ⎡⎣ ⎤⎦ ∈ ξ , 1 2 3 , and in the case of maximal D 1 - asymmetry we obtain ⎡⎣ ⎤⎦ ∈ ζ , 1 1 3 4 and ⎤ ⎦ ( ∈ ξ , 1 1 2 , implying that maximal asymmetry implies a very high degree of dependence in both cases. Furthermore, we study various topological proper ties of the family of copulas with maximal D 1 - asymmetry and derive some surprising properties for maximal D p - asymmetric copulas.

, holds for all x and y. The study of exchangeable random variables has exhibited a lot of interest in statistics (see, for instance, [8] and references therein). In case X and Y are identically distributed and have distribution function F, then ( ) X Y , is exchangeable if and only if the underlying copula A coincides with its transpose A t (defined as Hence, in what follows we consider continuous and identically distributed random variables X and Y . While the class of continuous exchangeable random variables X and Y is uniquely characterized by the class of symmetric copulas, the exact opposite, i.e., maximal non-exchangeability of random variables, strongly depends on the choice of measure quantifying the degree of non-exchangeability. One natural measure of non-exchangeability was studied by Nelsen [21] as well as by Klement and Mesiar [15], who independently showed that , t . Moreover, they characterized all copulas ∈ A with maximal ∞ d -asymmetry and showed that these copulas always model slightly negatively correlated random variables X and Y in the sense of Spearman's ρ. More precisely, ( ) = δ A 1 implies ( ) ⎡ ⎣ ⎤ ⎦ ∈ − − ρ A , 5 9 1 3 . Similar results also hold for different measures of concordance (see [17]).
Considering other metrics on the space of copulas yields alternative measures of non-exchangeability ( [13,25]): In [13] the stronger conditioning-based metric D 1 introduced in [27] was studied and the authors proved (among other things) that every copula ∈ A with maximal D 1 -asymmetry (i.e., ( 2 ) is not maximal asymmetric with respect to ∞ d and that no maximal ∞ d -asymmetric copula is maximal asymmetric with respect to D 1 .
Building upon the results in [13] we here further investigate the family of copulas with maximal D 1 -asymmetry, derive additional novel characterizations in terms of the Markov-product of copulas (see [3]), and study various topological properties; inter alia we prove that the family of mutually completely dependent copulas with maximal D 1 -asymmetry is dense in the set of all copulas with maximal D 1 -asymmetry. Furthermore, we extend the concept of maximal D 1 -asymmetry to the general D p -metrics ( , 0 , , 0 , d d , denote the Markov kernels (regular conditional distributions) of ∈ A B , , respectively. Although all D p -metrics induce the same topology, we show the surprising result that maximal D 1 -asymmetry is not equivalent to maximal D p -asymmetry for ( ) ∈ ∞ p 1, . In fact, copulas with maximal D p -asymmetry with ( ) ∈ ∞ p 1, are always mutually completely dependent and maximal asymmetric w.r.t. D 1 . Moreover, we tackle the question on the degree of dependence of copulas exhibiting maximal asymmetry with respect to ∞ d or D p for every [ ] ∈ ∞ p 1, . Since measures of concordance are generally not suitable for quantifying dependence (see, for instance, [11]) we consider the dependence measures ζ 1 introduced in [27] and further studied in [10,11], as well as ξ , defined in [4] and reinvestigated in [2]. Both measures have recently attracted a lot of interest (see, e.g., [1,10,11,14,24,26]) since, in contrast to standard methods like Spearman's ρ or Kendall's τ, these measures are 1 if and only if Y is a function of X and 0 if and only if X and Y are independent; moreover, they can be estimated consistently without underlying smoothness assumptions. We prove that when considering maximal ∞ d -asymmetry ⎡ ⎣ ⎤ ⎦ ∈ ζ , 1 1 5 6 and ⎡ ⎣ ⎤ ⎦ ∈ ξ , 1 2 3 hold, and in the case of maximal D 1 -asymmetry ⎡ ⎣ ⎤ ⎦ ∈ ζ , 1 1 3 4 and ⎤ ⎦ ( ∈ ξ , 1 1 2 follows. In other words, maximal nonexchangeable random variables (in the sense of ∞ d or D p ) always imply a high degree of dependence w.r.t. ζ 1 and ξ .
The rest of this article is organized as follows: Section 2 gathers preliminaries and notations that will be used throughout the article. In Section 3, we study possible values of ζ 1 and ξ for maximal ∞ d -asymmetric copulas and discuss an example illustrating differences of ζ 1 and ξ in the context of ordinal sums. In Section 4, we revisit copulas with maximal D 1 -asymmetry and derive several topological properties. Extensions on maximal D p -asymmetry for [ ] ∈ ∞ p 1, and some interrelations are established in Section 5. Consequences on the dependence measures ζ 1 and ξ conclude the article (Section 6). Various examples and graphics illustrate both the obtained results and the ideas underlying the proofs.

Notation and preliminaries
For every metric space ( ) Ω d , the Borel σ-field in Ω will be denoted by ( ) Ω , λ will denote the Lebesgue measure on ( ).
will denote the class of all measurable λ-preserving transformations on [ ] 0, 1 , i.e., and b the subclass of all bijective ∈ T . Throughout the article will denote the family of all twodimensional copulas, the family of all doubly-stochastic measures (for background on copulas and doubly stochastic measures we refer to [6,22] and references therein). Furthermore, M denotes the upper Fréchet Hoeffding bound, Π the product copula, and W the lower Fréchet Hoeffding bound. Additionally, the completely dependent copula induced by a measure-preserving transformation ∈ h will be denoted by C h (see [27], Definition 9). The family of all completely dependent copulas will be denoted by cd and the family of all mutually completely dependent copulas by For every copula ∈ C the corresponding doubly stochastic measure will be denoted by μ C . As usual, ∞ d denotes the uniform metric on , i.e., is a compact metric space (see [6]). In what follows, Markov kernels will play an important role. A mapping and the mapping ( holds -a.s. It is well-known that a regular conditional distribution of Y given X exists and is unique X -almost sure (where X denotes the distribution of X, i.e., the push-forward of via X). For every ∈ A (a version of) the corresponding regular conditional distribution (i.e., the regular conditional distribution of Y given X in the case that ( ) For more details and properties of conditional expectations and regular conditional distributions we refer to [12,16]. Expressing copulas in terms of their corresponding regular conditional distribution yields metrics stronger than ∞ d (see [27]) and defined by To simplify notation we will also write ( ) . We will also work with ∂ D , defined by whereby A t denotes the transpose of ∈ A . The metric ∂ D can be seen as metrization of the so-called ∂-convergence, introduced and studied in [18,19]. In [27], it is shown that ( ) D , 1 is a complete and separable metric space with diameter 1/2 and that the topology induced by D 1 is strictly finer than the one induced by ∞ d . For further background on D 1 and ∂ D as well as for possible extensions to the multivariate setting we refer to [6,7,10,27] and references therein.
The D 1 -based dependence measure ζ 1 (introduced in [27] and further investigated in [10,11]) is defined as In the sequel, we will also consider the dependence measure ξ (first introduced in [4] and reinvestigated in [2]) defined as where μ is the law of Y . In the copula setting, it is straightforward to verify that ξ can be expressed in terms , 0 ,1 (see, e.g., [5,9]), the following simple result holds: Proof. According to Lemma 3.1 in [9] for ∈ h b the Markov kernel of ( ) A h can be expressed as  In the sequel, we will also work with rearrangements [23] (see [26] for an elegant application of rearrangements in the copula context). We call [ ] → * f : 0, 1 the decreasing rearrangement of a Borel measurable function ,0 , d , whereby the rearrangement is applied on the first coordinate of ( ) In [26], it was shown that ↑ A is an SI copula and both dependence measures ζ 1 The family of all SI copulas will be denoted by ↑ . For further information we refer to [22] and references therein. Given , a new copula denoted by * A B can be constructed via the so-called star/Markov product * A B (see [3]) by denotes the partial derivative of A with respect to the first coordinate. The star product * A B is always a copula, i.e., no smoothness assumptions on A B , are required. Translating to the Markov kernel setting the star product corresponds to the well-known composition of Markov kernels and the following lemma holds: , and let K K , A B denote the Markov kernels of A and B, respectively. Then the Markov kernel is a regular conditional distribution of * A B.
3 Maximal d ∞ -asymmetric copulas and their extent of dependence with respect to ζ 1 and ξ Since ordinal sums will play an important role in what follows, we briefly recall their definition. We follow [6] and let ⊆ I be some finite index set, (( )) ∈ a b , i i i I be a family of non-overlapping intervals with holds. Furthermore, ( ) ∈ C i i I denotes a family of bivariate copulas. Then the copula C defined by is an ordinal sum, and we write The following lemma gathers some useful formulas for D 1 and D 2 2 , which will be used in the sequel.
as in the theorem. □ As a direct consequence, the dependence measure ξ of ordinal sums can easily be expressed in terms of ( ) ξ C i : , , , , , holds, where f is defined according to Lemma 3.1 and only depends on the partition.
The following example shows that ordinal sums can be used to construct copulas attaining every possible dependence value w.r.t. to ξ and ζ 1 . Figure 1 depicts the support of μ Cs for different choices of [ ] ∈ s 0, 1 . Using . Therefore, the map 1, 2, 3 2 (and μ A f ij denotes the push-forward of μ A via f ij ).
Proof. We may assume that ( ) = A , . Then there exist copulas such that . Applying Lemmas 2.1 and 3.2 we therefore obtain , , , , , 0, 1 and using the same arguments as in Example 3.3 it follows that for every Since ζ 1 and ξ are similar by construction, one might expect the analogous statements for ζ 1 . Note, however, that a different proof is needed since according to Lemma 3.1 the formulas for D 1 are more involved.
. Furthermore, for every Proof. Proceeding as in the proof of Theorem 3.4 we obtain ( ) . Considering the is an ordinal sum again and can be expressed as holds pointwise. Due to the fact that ζ 1 is monotone w.r.t. the pointwise order on ↑ and ζ 1 is invariant w.r.t. to (SI)-rearrangements (see [26]), we obtain where the last equality follows from Lemma 3.1 (the detailed calculations are deferred to Appendix A). To show the second assertion we can proceed analogously to the proof of Theorem 3.4 and use shrunk copies of the copula C s defined in Example 3. (7), ζ 1 exhibits a different behavior as demonstrated in the following example: Then a version of the corresponding Markov kernel of A 1 is given by and let A denote the ordinal sum given by and C Π be the ordinal sum given by  and analogously we obtain Remark 3.7. Considering the monotonicity of ζ 1 with respect to the pointwise order on ↑ as proved in [26], Example 3.6 shows that there exist copulas

Maximal D 1 -asymmetry of copulas revisited
In this section, we complement characterizations of copulas with maximal D 1 -asymmetry going back to [13] and derive some topological properties of subclasses. To be consistent with the notation in [13], the family of copulas with maximal D 1 -asymmetry is denoted by the subclass of mutually completely dependent copulas is denoted by = mcd κ 1 . We start with the family of mutually completely dependent copulas and show closedness w.r.t. the metric ∂ D . .
Proof. Let ( ) ∈ A h n n be a sequence of mutually completely dependent copulas with ∂ D -limit A. Since according to [27] the family of completely dependent copulas is closed w.r.t. D 1 we obtain ∈ A cd and ∈ A t cd . Using [27, Lemma 10] there exist λ-preserving transformations ′ ∈ g g , such that a version of the Markov kernel ( ) respectively. Furthermore, since a copula A is completely dependent if and only if it is left-invertible w.r.t. the * -product and there exists some x y , (see [27]) we have  for every ∈ n . Applying [27][Proposition 15 (ii)] yields  Figure 1 in [9]). Setting  Leaving the subclass of mutually completely dependent copulas we will now derive novel and handy characterizations of copulas with maximal D 1 -asymmetry and then show some topological properties. The following lemma, showing that the * -product cannot increase the D p -distance, will be useful in the sequel. The result has already been stated for D 1 in a slightly different context in [28].    The next theorem gathers several equivalent characterizations of copulas having maximal D 1 -asymmetry (see [13]), and the novel ones established here are (v) and (vi).

Theorem 4.4. For every ∈
A the following statements are equivalent: 0, 1 with the following properties: such that the following identity holds, whereby ( ) (v) ( Proof. The equivalences of (i), (ii), (iii), and (iv) have already been proved in [13]. Note that the equivalence in property (ii) directly follows from the facts that Φ A A , t is Lipschitz continuous with Lipschitz constant 2 and the function [ is Lipschitz continuous with Lipschitz constant 2 (see [27] [Lemma 5]), the property that Not surprisingly, the following result holds.
n n t n t t n t and hence , , t n t n t Remark 4.7. Proposition 4.6 certainly is not surprising, however, the following result is. Key for proving the statement is property (v) of Theorem 4.4.
, , , 1 2 are copulas, that ( ) = κ A 1 n for every ∈ n , and that Since the * -product is jointly continuous w.r.t. D 1 (see [29]) we have and copulas ∈ C C C C , , , 1 2 3 4 such that It is well-known that mcd (in fact even the family of straight shuffles) is dense in ( ) (see, e.g., [6] [Corollary 4.1.16]), hence, we can find mutually completely dependent copulas and applying Theorem 4.4 we conclude that Ã has maximal D 1 -asymmetry too. Furthermore, using the triangle inequality we obtain x 0, 1 , which is equivalent to Ã being completely dependent (see [3]). Using the same arguments we also obtain for every x 0, 1 , i.e., Ã t is completely dependent too. Altogether, we have shown that ∈ Ã mcd , which completes the proof. □

Maximal D p -asymmetry
Since the metrics D p , [ ] ∈ ∞ p 1, induce the same topology on one could conjecture that maximal D p -asymmetry might be the same as maximal D 1 -asymmetry. We will falsify this idea and start with three simple lemmata. holds for every ( ) ∈ ∞ p 1, .
Lemma 5.2. The metric space ( ) D , p has the following diameter: Proof. According to Lemma 5 in [27] we have it is straightforward to verify that The assertion for = ∞ p is a direct consequence of Lemma 5 in [27]. □ Slightly adapting the notation of the previous section we will now focus on the family 1, also has maximal D 1 -asymmetry.
Proof. Using the inequality ( , we obtain The following example, however, shows that the reverse implication does not hold in general. Example 5.4. Suppose that ∈ A corresponds to the uniform distribution on the union of the four squares ( Figure 4) Since A (and A t ) is a checkerboard copula (see, for instance, [11,18] Figure 4: Density of the copula A (left panel) and the copula A t (right panel) considered in Example 5.4. The copula A has maximal D 1 -asymmetry, i.e., κ A It is straightforward to verify ( ) = κ A 1 1 (e.g., by using property (iv) or property (v) in Theorem 4.4). On the other hand, simple calculations (see Appendix A) yield . As a direct consequence we obtain ( 1, , i.e., although A has maximal D 1 -asymmetry, it fails to have maximal D p -asymmetry.
Contrary to D 1 , the class only contains mutually completely dependent copulas.
, then A is a mutually completely dependent copula. Corollary 5.6. The following properties hold: 6 Maximal D p -asymmetric copulas and their values for ζ 1 and ξ In Section 3, we have shown that copulas ∈ A with maximal ∞ d -asymmetry have very high dependence scores with respect to ζ 1 and ξ . Here we now focus on the range of these dependence measures to maximal D p -asymmetric copulas.
and ( ) ξ A are 1 if and only if A is completely dependent, the assertion directly follows from Theorem 5.5. □ For the case = p 1 different values for ξ and ζ 1 are possible.

holds.
Proof. Proceeding analogously to the proof of Theorem 4.4 we obtain ( On the other hand, holds.
Proof. Using the same arguments as in the proof of Theorem 6.2 we find copulas holds. Since Ã is an ordinal sum it is clear that the (SI)-rearrange- . Hence, using Lemma 4.3 we obtain whereby we used the fact that ( ) D A, Π 1 is monotone with respect to the pointwise order in ↑ (see [26] which completes the proof. □ The following example demonstrates that it is possible to find copulas ∈ = A κ 1 1 such that ( ) ζ A 1 (or ( ) ξ A , respectively) is arbitrarily close to the lower bound derived in Theorems 6.2 and 6.3.
be a natural number with ≥ n 3, set ≔ N 2 n and define the sets  Obviously, we have ( ) the copula corresponding to the uniform distribution on the union of the four sets    Remark 6.5. Slightly modifying the construction from Example 6.4 (which corresponds to copying shrunk versions of the product copula Π in the small squares) we now construct the copula B N by copying shrunk versions of M in every square of the "diagonal" of each of the four sets × U V we can find a copula C N α with ( ) = ζ C s N α 1 and the same result holds for ζ 1 replaced by ξ . In other words, each point in the intervals mentioned in Theorems 6.2 and 6.3 is attained.