In this article, we focus on copulas underlying maximal non-exchangeable pairs of continuous random variables either in the sense of the uniform metric or the conditioning-based metrics , and analyze their possible extent of dependence quantified by the recently introduced dependence measures and . Considering maximal -asymmetry we obtain and , and in the case of maximal -asymmetry we obtain and , implying that maximal asymmetry implies a very high degree of dependence in both cases. Furthermore, we study various topological properties of the family of copulas with maximal -asymmetry and derive some surprising properties for maximal -asymmetric copulas.
Two random variables and with joint distribution function are called exchangeable if and only if the pairs and have the same distribution, or equivalently, if holds for all and . The study of exchangeable random variables has exhibited a lot of interest in statistics (see, for instance,  and references therein). In case and are identically distributed and have distribution function , then is exchangeable if and only if the underlying copula coincides with its transpose (defined as ). Hence, in what follows we consider continuous and identically distributed random variables and . While the class of continuous exchangeable random variables and 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  as well as by Klement and Mesiar , who independently showed that
holds for every and introduced the -based measure via . Moreover, they characterized all copulas with maximal -asymmetry and showed that these copulas always model slightly negatively correlated random variables and in the sense of Spearman’s . More precisely, implies . Similar results also hold for different measures of concordance (see ).
Considering other metrics on the space of copulas yields alternative measures of non-exchangeability ([13,25]): In  the stronger conditioning-based metric introduced in  was studied and the authors proved (among other things) that every copula with maximal -asymmetry (i.e., ) is not maximal asymmetric with respect to and that no maximal -asymmetric copula is maximal asymmetric with respect to .
Building upon the results in  we here further investigate the family of copulas with maximal -asymmetry, derive additional novel characterizations in terms of the Markov-product of copulas (see ), and study various topological properties; inter alia we prove that the family of mutually completely dependent copulas with maximal -asymmetry is dense in the set of all copulas with maximal -asymmetry. Furthermore, we extend the concept of maximal -asymmetry to the general -metrics ( ), defined by
where denote the Markov kernels (regular conditional distributions) of , respectively. Although all -metrics induce the same topology, we show the surprising result that maximal -asymmetry is not equivalent to maximal -asymmetry for . In fact, copulas with maximal -asymmetry with are always mutually completely dependent and maximal asymmetric w.r.t. .
Moreover, we tackle the question on the degree of dependence of copulas exhibiting maximal asymmetry with respect to or for every . Since measures of concordance are generally not suitable for quantifying dependence (see, for instance, ) we consider the dependence measures introduced in  and further studied in [10,11], as well as , defined in  and reinvestigated in . 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 or Kendall’s , these measures are 1 if and only if is a function of and 0 if and only if and are independent; moreover, they can be estimated consistently without underlying smoothness assumptions. We prove that when considering maximal -asymmetry and hold, and in the case of maximal -asymmetry and follows. In other words, maximal non-exchangeable random variables (in the sense of or ) always imply a high degree of dependence w.r.t. and .
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 and for maximal -asymmetric copulas and discuss an example illustrating differences of and in the context of ordinal sums. In Section 4, we revisit copulas with maximal -asymmetry and derive several topological properties. Extensions on maximal -asymmetry for and some interrelations are established in Section 5. Consequences on the dependence measures and conclude the article (Section 6). Various examples and graphics illustrate both the obtained results and the ideas underlying the proofs.
2 Notation and preliminaries
For every metric space the Borel -field in will be denoted by , will denote the Lebesgue measure on . will denote the class of all measurable -preserving transformations on , i.e.,
and the subclass of all bijective . Throughout the article will denote the family of all two-dimensional 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, denotes the upper Fréchet Hoeffding bound, the product copula, and the lower Fréchet Hoeffding bound. Additionally, the completely dependent copula induced by a measure-preserving transformation will be denoted by (see , Definition 9). The family of all completely dependent copulas will be denoted by and the family of all mutually completely dependent copulas by . For every copula the corresponding doubly stochastic measure will be denoted by . As usual, denotes the uniform metric on , i.e.,
for every . It is well-known that is a compact metric space (see ).
In what follows, Markov kernels will play an important role. A mapping is called a Markov kernel from to if the mapping is measurable for every fixed and the mapping is a probability measure for every fixed . A Markov kernel is called regular conditional distribution of a (real-valued) random variable given (another random variable) if for every
holds -a.s. It is well-known that a regular conditional distribution of given exists and is unique -almost sure (where denotes the distribution of , i.e., the push-forward of via ). For every (a version of) the corresponding regular conditional distribution (i.e., the regular conditional distribution of given in the case that ) will be denoted by . Note that for every and Borel sets we have
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 (see ) and defined by
To simplify notation we will also write . We will also work with , defined by
whereby denotes the transpose of . The metric can be seen as metrization of the so-called -convergence, introduced and studied in [18,19]. In , it is shown that is a complete and separable metric space with diameter 1/2 and that the topology induced by is strictly finer than the one induced by . For further background on and as well as for possible extensions to the multivariate setting we refer to [6,7, 10,27] and references therein.
where is the law of . In the copula setting, it is straightforward to verify that can be expressed in terms of and that holds. Both dependence measures attain values in and are 0 if and only if , and 1 if and only if is completely dependent.
Letting denote the generalized shuffle of w.r.t. the first coordinate, implicitly defined via the corresponding doubly stochastic measure by
Let be a -preserving bijection. Then and hold for every .
According to Lemma 3.1 in  for the Markov kernel of can be expressed as and for we obtain
which proves the assertion.□
In the sequel, we will also work with rearrangements  (see  for an elegant application of rearrangements in the copula context). We call the decreasing rearrangement of a Borel measurable function if it fulfills . The stochastically increasing (SI)-rearrangement of is then defined as
whereby the rearrangement is applied on the first coordinate of , i.e., for every fixed the rearranged Markov kernel is defined via . In , it was shown that is an SI copula and both dependence measures and are invariant w.r.t. to the rearrangement, i.e., they fulfill and , respectively. Recall that a copula is called SI if there exists a Borel set with such that for any the mapping is non-increasing on . The family of all SI copulas will be denoted by . For further information we refer to  and references therein.
Given a new copula denoted by can be constructed via the so-called star/Markov product (see ) by
where denotes the partial derivative of with respect to the first coordinate. The star product is always a copula, i.e., no smoothness assumptions on 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:
 Suppose that and let denote the Markov kernels of A and B, respectively. Then the Markov kernel , defined by
is a regular conditional distribution of .
3 Maximal -asymmetric copulas and their extent of dependence with respect to and
Since ordinal sums will play an important role in what follows, we briefly recall their definition. We follow  and let be some finite index set, be a family of non-overlapping intervals with for each such that holds. Furthermore, denotes a family of bivariate copulas. Then the copula defined by
is an ordinal sum, and we write . The following lemma gathers some useful formulas for and , which will be used in the sequel.
Let be an ordinal sum with for some . Then
whereby f and g are given by and , respectively.
The definition of yields
for every . Using the fact that (without loss of generality) the Markov kernel of is 0 below the squares and 1 above , and applying change of coordinates yields
Analogously, we obtain
with as in the theorem.□
As a direct consequence, the dependence measure of ordinal sums can easily be expressed in terms of :
Let be an ordinal sum with for some . Then
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 and .
Consider , whereby , , and for and set as well as . Figure 1 depicts the support of for different choices of . Using Corollary 3.2 we have . Therefore, the map defined by is continuous and onto. The same holds for .
Before deriving some first results concerning the range of the dependence measures and for maximal -asymmetric copulas , we recall the characterizations of maximal -asymmetry derived in [21,15]: is maximal if and only if and or and . Without loss of generality we may focus on the case and . Since is doubly stochastic in this case we obviously have . As a direct consequence, we can find copulas fulfilling
whereby the functions are given by for each (and denotes the push-forward of via ).
If has maximal -asymmetry, i.e., if holds, then satisfies . Moreover, for every there exists a copula A with fulfilling .
We may assume that and . Then there exist copulas such that holds. Defining by
with equality if and only if for every .
Defining by with as in Example 3.3 yields
Considering for and using the same arguments as in Example 3.3 it follows that for every we find a copula with and .□
Since and are similar by construction, one might expect the analogous statements for . Note, however, that a different proof is needed since according to Lemma 3.1 the formulas for are more involved.
If has maximal -asymmetry, then satisfies . Furthermore, for every there exists a copula with fulfilling .
Proceeding as in the proof of Theorem 3.4 we obtain . Considering the (SI)-rearrangement of it is clear that is an ordinal sum again and can be expressed as . Since every is SI and hence fulfills for every and every (see, e.g., [Section 5.2]), we obtain that
holds pointwise. Due to the fact that is monotone w.r.t. the pointwise order on and is invariant w.r.t. to (SI)-rearrangements (see ), 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 defined in Example 3.3 (see Appendix A).□
While the minimum value of for a copula with maximal -asymmetry is attained if and only if for every in equation (7), exhibits a different behavior as demonstrated in the following example:
Let be defined by
Then a version of the corresponding Markov kernel of is given by . Furthermore, we set and and let denote the ordinal sum given by and be the ordinal sum given by (Figure 2).
By construction we have , however, considering
and analogously we obtain
Applying Lemma 3.1 we obtain .
4 Maximal -asymmetry of copulas revisited
In this section, we complement characterizations of copulas with maximal -asymmetry going back to  and derive some topological properties of subclasses. To be consistent with the notation in , the family of copulas with maximal -asymmetry is denoted by
the subclass of mutually completely dependent copulas is denoted by . We start with the family of mutually completely dependent copulas and show closedness w.r.t. the metric .
The set is closed in .
Let be a sequence of mutually completely dependent copulas with -limit . Since according to  the family of completely dependent copulas is closed w.r.t. we obtain and . Using [27, Lemma 10] there exist -preserving transformations such that a version of the Markov kernel and is given by and , respectively. Furthermore, since a copula is completely dependent if and only if it is left-invertible w.r.t. the -product (see ) we have . Applying Lemma 2.2 therefore yields that for -a.e. . Using the fact that is surjective -almost everywhere, there exists a -preserving and bijective transformation such that holds -a.e., implying . It remains to show that , which can be done as follows. Using [Theorem 3.5] and the triangle inequality we obtain
for every . Applying [Proposition 15 (ii)] yields
Together with the fact that the maximal distance cannot exceed it follows that , which completes the proof.□
The following example shows that the set is not closed w.r.t. the metric .
Let be the mutually completely dependent copula induced by the bijective measure-preserving transformation , given by
for all and let be the completely dependent copula induced by the -preserving transformation given by (see Figure 1 in ). Setting and we have and and according to [9, Example 3.3] it is straightforward to verify that . As a next step, we reorder the shrunk copulas to obtain maximal -asymmetry. Let denote the -preserving interval exchange transformation defined by and, furthermore, let and denote the respective shuffles (Figure 3). Due to the fact that the metric is shuffle-invariant w.r.t. bijective transformations (using the same arguments as in the proof of Lemma 2.1) yields
Setting and considering property (3) of Theorem 4.1 in  (see also Theorem 4.4 (iii) in the sequel) we directly obtain that and are maximal asymmetric w.r.t. , which shows that is not closed w.r.t. the metric .
Leaving the subclass of mutually completely dependent copulas we will now derive novel and handy characterizations of copulas with maximal -asymmetry and then show some topological properties. The following lemma, showing that the -product cannot increase the -distance, will be useful in the sequel. The result has already been stated for in a slightly different context in .
For every , the following inequality holds for every :
Applying Lemma 2.2, Jensen’s inequality, disintegration and using the fact that is doubly stochastic we obtain
which completes the proof.□
The next theorem gathers several equivalent characterizations of copulas having maximal -asymmetry (see ), and the novel ones established here are (v) and (vi).
For every the following statements are equivalent:
(or equivalently, has maximal -asymmetry),
there exists a Borel set with the following properties:
there exist sets with , , and and , and copulas such that the following identity