Maximal asymmetry of bivariate copulas and consequences to measures of dependence

1 Introduction

Two random variables X and Y with joint distribution function H are called exchangeable if and only if the pairs ( X , Y ) and ( Y , X ) have the same distribution, or equivalently, if H ( x , y ) = H ( y , x ) 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 A t ( x , y ) = A ( y , x ) ). 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

holds for every A ∈ C and introduced the d ∞ -based measure δ : C → [ 0 , 1 ] via δ ( A ) ≔ 3 d ∞ ( A , A t ) . Moreover, they characterized all copulas A ∈ C 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 ∈ C with maximal D 1 -asymmetry (i.e., D 1 ( A , A t ) = 1 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 ( p ∈ [ 1 , ∞ ) ), defined by

where K A ( ⋅ , ⋅ ) , K B ( ⋅ , ⋅ ) denote the Markov kernels (regular conditional distributions) of A , B ∈ C , 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 ∈ 5 6 , 1 and ξ ∈ 2 3 , 1 hold, and in the case of maximal D 1 -asymmetry ζ 1 ∈ 3 4 , 1 and ξ ∈ 1 2 , 1 follows. In other words, maximal non-exchangeable 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.

2 Notation and preliminaries

For every metric space ( Ω , d ) the Borel σ -field in Ω will be denoted by ℬ ( Ω ) , λ will denote the Lebesgue measure on ℬ ( R ) . T will denote the class of all measurable λ -preserving transformations on [ 0 , 1 ] , i.e.,

and T b the subclass of all bijective T ∈ T . Throughout the article C will denote the family of all two-dimensional copulas, P 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 ∈ T will be denoted by C h (see [27], Definition 9). The family of all completely dependent copulas will be denoted by C c d and the family of all mutually completely dependent copulas by C m c d ≔ { C h ∈ C c d : h ∈ T b } . For every copula C ∈ C the corresponding doubly stochastic measure will be denoted by μ C . As usual, d ∞ denotes the uniform metric on C , i.e.,

for every A , B ∈ C . It is well-known that ( C , d ∞ ) is a compact metric space (see [6]).

In what follows, Markov kernels will play an important role. A mapping K : R × ℬ ( R ) → [ 0 , 1 ] is called a Markov kernel from ( R , ℬ ( R ) ) to ( R , ℬ ( R ) ) if the mapping x ↦ K ( x , B ) is measurable for every fixed B ∈ ℬ ( R ) and the mapping B ↦ K ( x , B ) is a probability measure for every fixed x ∈ R . A Markov kernel K : R × ℬ ( R ) → [ 0 , 1 ] is called regular conditional distribution of a (real-valued) random variable Y given (another random variable) X if for every B ∈ ℬ ( R )

holds P -a.s. It is well-known that a regular conditional distribution of Y given X exists and is unique P X -almost sure (where P X denotes the distribution of X , i.e., the push-forward of P via X ). For every A ∈ C (a version of) the corresponding regular conditional distribution (i.e., the regular conditional distribution of Y given X in the case that ( X , Y ) ∼ A ) will be denoted by K A ( ⋅ , ⋅ ) . Note that for every A ∈ C and Borel sets E , F ∈ ℬ ( [ 0 , 1 ] ) 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 d ∞ (see [27]) and defined by

To simplify notation we will also write Φ A , B ( y ) ≔ ∫ [ 0 , 1 ] ∣ K A ( x , [ 0 , y ] ) − K B ( x , [ 0 , y ] ) ∣ d λ ( x ) . We will also work with D ∂ , defined by

whereby A t denotes the transpose of A ∈ C . 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 ( C , 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

whereby ( X , Y ) has copula A ∈ C . 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 of D 2 and that ξ ( X , Y ) ≔ ξ ( A ) = 6 D 2 2 ( A , Π ) holds. Both dependence measures attain values in [ 0 , 1 ] and are 0 if and only if A = Π , and 1 if and only if A is completely dependent.

Letting S h ( A ) denote the generalized shuffle of A w.r.t. the first coordinate, implicitly defined via the corresponding doubly stochastic measure μ A by

for all E , F ∈ ℬ ( [ 0 , 1 ] ) (see, e.g., [5,9]), the following simple result holds:

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 ] → R the decreasing rearrangement of a Borel measurable function f : [ 0 , 1 ] → R if it fulfills f ∗ ( t ) ≔ inf { x ∈ R : λ ( { z ∈ [ 0 , 1 ] : f ( z ) > x } ) ≤ t } . The stochastically increasing (SI)-rearrangement A ↑ of A is then defined as

whereby the rearrangement is applied on the first coordinate of K A ( ⋅ , ⋅ ) , i.e., for every fixed y ∈ [ 0 , 1 ] the rearranged Markov kernel is defined via K A ( t , [ 0 , y ] ) ∗ ≔ inf { x ∈ [ 0 , 1 ] : λ ( { z ∈ [ 0 , 1 ] : K A ( z , [ 0 , y ] ) > x } ) ≤ t } . In [26], it was shown that A ↑ is an SI copula and both dependence measures ζ 1 and ξ are invariant w.r.t. to the rearrangement, i.e., they fulfill ζ 1 ( A ↑ ) = ζ 1 ( A ) and ξ ( A ↑ ) = ξ ( A ) , respectively. Recall that a copula A is called SI if there exists a Borel set Λ ⊆ [ 0 , 1 ] with λ ( Λ ) = 1 such that for any y ∈ [ 0 , 1 ] the mapping x ↦ K A ( x , [ 0 , y ] ) is non-increasing on Λ . The family of all SI copulas will be denoted by C ↑ . For further information we refer to [22] and references therein.

Given A , B ∈ C a new copula denoted by A ∗ B can be constructed via the so-called star/Markov product A ∗ B (see [3]) by

where ∂ 1 A ( x , y ) 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:

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 ⊆ N be some finite index set, ( ( a i , b i ) ) i ∈ I be a family of non-overlapping intervals with 0 ≤ a i < b i ≤ 1 for each i ∈ I such that ⋃ i ∈ I [ a i , b i ] = [ 0 , 1 ] 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 C = ( ⟨ a i , b i , C i ⟩ ) i ∈ I . The following lemma gathers some useful formulas for D 1 and D 2 2 , which will be used in the sequel.

As a direct consequence, the dependence measure ξ of ordinal sums can easily be expressed in terms of ξ ( C i ) :

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

Mass distribution of the doubly stochastic measure μ C s for s = 0.3 (left panel) and s = 0.8 (right panel) considered in Example 3.3. For the dependence measures ξ and ζ 1 we obtain ξ ( C 0.3 ) = 0.91 and ξ ( C 0.8 ) = 0.36 as well as ζ 1 ( C 0.3 ) = 0.973 and ζ 1 ( C 0.8 ) = 0.488 .

Before deriving some first results concerning the range of the dependence measures ξ ( A ) and ζ 1 ( A ) for maximal d ∞ -asymmetric copulas A , we recall the characterizations of maximal d ∞ -asymmetry derived in [21,15]: d ∞ ( A , A t ) is maximal if and only if A 2 3 , 1 3 = 0 and A 1 3 , 2 3 = 1 3 or A t 2 3 , 1 3 = 0 and A t 1 3 , 2 3 = 1 3 . Without loss of generality we may focus on the case A 1 3 , 2 3 = 1 3 and A 2 3 , 1 3 = 0 . Since A is doubly stochastic in this case we obviously have μ A 0 , 1 3 × 1 3 , 2 3 = μ A 1 3 , 2 3 × 2 3 , 1 = μ A 2 3 , 1 × 0 , 1 3 = 1 3 . As a direct consequence, we can find copulas A 1 , A 2 , A 3 ∈ C fulfilling

whereby the functions f i j : [ 0 , 1 ] 2 → i − 1 3 , i 3 × j − 1 3 , j 3 are given by f i j ( x , y ) = x + i − 1 3 , y + j − 1 3 for each ( i , j ) ∈ { 1 , 2 , 3 } 2 (and μ A f i j denotes the push-forward of μ A via f i j ).