Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access August 24, 2022

Maximal asymmetry of bivariate copulas and consequences to measures of dependence

Florian Griessenberger and Wolfgang Trutschnig EMAIL logo
From the journal Dependence Modeling

Abstract

In this article, we focus on copulas underlying maximal non-exchangeable pairs ( X , Y ) of continuous 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 quantified by the recently introduced dependence measures ζ 1 and ξ . Considering maximal d -asymmetry we obtain ζ 1 5 6 , 1 and ξ 2 3 , 1 , and in the case of maximal D 1 -asymmetry we obtain ζ 1 3 4 , 1 and ξ 1 2 , 1 , 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 D 1 -asymmetry and derive some surprising properties for maximal D p -asymmetric copulas.

MSC 2010: 62H05; 62H20; 60E05; 54E52

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

d ( A , A t ) sup x , y [ 0 , 1 ] A ( x , y ) A ( y , x ) 1 3

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

(1) D p ( A , B ) [ 0 , 1 ] [ 0 , 1 ] K A ( x , [ 0 , y ] ) K B ( x , [ 0 , y ] ) p d λ ( x ) d λ ( y ) 1 p ,

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 ( Ω ) , λ will denote the Lebesgue measure on ( R ) . T will denote the class of all measurable λ -preserving transformations on [ 0 , 1 ] , i.e.,

T = { T : [ 0 , 1 ] [ 0 , 1 ] measurable with λ ( T 1 ( E ) ) = λ ( E ) E ( [ 0 , 1 ] ) } ,

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.,

d ( A , B ) max ( x , y ) [ 0 , 1 ] 2 A ( x , y ) B ( x , y )

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 × ( R ) [ 0 , 1 ] is called a Markov kernel from ( R , ( R ) ) to ( R , ( R ) ) if the mapping x K ( x , B ) is measurable for every fixed B ( R ) and the mapping B K ( x , B ) is a probability measure for every fixed x R . A Markov kernel K : R × ( R ) [ 0 , 1 ] is called regular conditional distribution of a (real-valued) random variable Y given (another random variable) X if for every B ( R )

K ( X ( ω ) , B ) = E ( 1 B Y X ) ( ω )

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 ( [ 0 , 1 ] ) we have

(2) E K A ( x , F ) d λ ( x ) = μ A ( E × F ) and [ 0 , 1 ] K A ( x , F ) d λ ( x ) = λ ( F ) .

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

(3) D p ( A , B ) [ 0 , 1 ] [ 0 , 1 ] K A ( x , [ 0 , y ] ) K B ( x , [ 0 , y ] ) p d λ ( x ) d λ ( y ) 1 p ,

(4) D ( A , B ) sup y [ 0 , 1 ] [ 0 , 1 ] K A ( x , [ 0 , y ] ) K B ( x , [ 0 , y ] ) d λ ( x ) .

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

D ( A , B ) D 1 ( A , B ) + D 1 ( A t , B t ) ,

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

ζ 1 ( X , Y ) ζ 1 ( A ) 3 D 1 ( A , Π ) ,

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

ξ ( X , Y ) Var ( E ( 1 { Y t } X ) ) d μ ( t ) Var ( 1 { Y t } ) d μ ( t ) ,

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

μ S h ( A ) ( E × F ) μ A ( h 1 ( E ) × F ) ,

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

Lemma 2.1

Let h T b be a λ -preserving bijection. Then ζ 1 ( S h ( A ) ) = ζ 1 ( A ) and ξ ( S h ( A ) ) = ξ ( A ) hold for every A C .

Proof

According to Lemma 3.1 in [9] for h T b the Markov kernel of S h ( A ) can be expressed as K S h ( A ) ( x , [ 0 , y ] ) = K A ( h 1 ( x ) , [ 0 , y ] ) and for p [ 1 , ) we obtain

D p p ( S h ( A ) , Π ) = [ 0 , 1 ] [ 0 , 1 ] K S h ( A ) ( x , [ 0 , y ] ) y p d λ ( x ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] K A ( h 1 ( x ) , [ 0 , y ] ) y p d λ ( x ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] K A ( h 1 ( x ) , [ 0 , y ] ) y p d λ h ( x ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] K A ( h 1 ( h ( x ) ) , [ 0 , y ] ) y p d λ ( x ) d λ ( y ) = D p p ( A , Π ) ,

which proves the assertion.□

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

A ( x , y ) [ 0 , x ] K A ( t , [ 0 , y ] ) d λ ( t ) ,

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

(5) ( A B ) ( x , y ) [ 0 , 1 ] 2 A ( x , t ) 1 B ( t , y ) d λ ( t ) ,

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:

Lemma 2.2

[29] Suppose that A , B C and let K A , K B denote the Markov kernels of A and B, respectively. Then the Markov kernel K A K B , defined by

(6) ( K A K B ) ( x , F ) [ 0 , 1 ] K B ( y , F ) K A ( x , d y ) ,

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 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

C ( x , y ) = a i + ( b i a i ) C i x a i b i a i , y a i b i a i , ( x , y ) ( a i , b i ) 2 min { x , y } elsewhere

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.

Lemma 3.1

Let C = ( a i , b i , C i ) i I be an ordinal sum with I { 1 , , n } for some n N . Then

D 2 2 ( C , Π ) = i = 1 n ( ( b i a i ) 2 D 2 2 ( C i , Π ) ) + f ( a 1 , , a n , b 1 , , b n ) , D 1 ( C , Π ) = i = 1 n ( b i a i ) 2 [ 0 , 1 ] [ 0 , 1 ] K C i ( x , [ 0 , y ] ) ( a i + ( b i a i ) y ) d λ ( x ) d λ ( y ) + g ( a 1 , , a n , b 1 , , b n ) ,

whereby f and g are given by f ( a 1 , , a n , b 1 , , b n ) i = 1 n ( b i a i ) 2 3 + ( b i a i ) ( 1 b i ) 1 3 and g ( a 1 , , a n , b 1 , , b n ) i = 1 n ( b i a i ) 1 2 b i + b i 2 2 + a i 2 2 , respectively.

Proof

The definition of D 2 2 yields

D 2 2 ( C , Π ) = [ 0 , 1 ] [ 0 , 1 ] ( K C ( x , [ 0 , y ] ) K Π ( x , [ 0 , y ] ) ) 2 d λ ( x ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] K C ( x , [ 0 , y ] ) 2 d λ ( x ) d λ ( y ) 2 [ 0 , 1 ] y [ 0 , 1 ] K C ( x , [ 0 , y ] ) d λ ( x ) d λ ( y ) + [ 0 , 1 ] [ 0 , 1 ] y 2 d λ ( x ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] K C ( x , [ 0 , y ] ) 2 d λ ( x ) d λ ( y ) 1 3

for every C C . Using the fact that (without loss of generality) the Markov kernel K C ( x , [ 0 , y ] ) of C is 0 below the squares ( a i , b i ) 2 and 1 above ( a i , b i ) 2 , and applying change of coordinates yields

D 2 2 ( C , Π ) + 1 3 = i = 1 n ( a i , b i ) ( a i , b i ) K C ( x , [ 0 , y ] ) 2 d λ ( x ) d λ ( y ) + ( b i , 1 ) ( a i , b i ) 1 d λ ( x ) d λ ( y ) = i = 1 n ( b i a i ) 2 [ 0 , 1 ] [ 0 , 1 ] K C i ( x , [ 0 , y ] ) 2 d λ ( x ) d λ ( y ) + ( b i a i ) ( 1 b i ) = i = 1 n ( b i a i ) 2 D 2 2 ( C i , Π ) + 1 3 + ( b i a i ) ( 1 b i ) .

Analogously, we obtain

D 1 ( C , Π ) = i = 1 n [ 0 , 1 ] ( a i , b i ) K C ( x , [ 0 , y ] ) y d λ ( x ) d λ ( y ) = i = 1 n ( a i , b i ) ( a i , b i ) K C i x a i b i a i , 0 , y a i b i a i y d λ ( x ) d λ ( y ) + i = 1 n ( b i , 1 ) ( a i , b i ) ( 1 y ) d λ ( x ) d λ ( y ) + i = 1 n ( 0 , a i ) ( a i , b i ) y d λ ( x ) d λ ( y ) = i = 1 n ( b i a i ) 2 [ 0 , 1 ] [ 0 , 1 ] K C i ( x , [ 0 , y ] ) ( a i + ( b i a i ) y ) d λ ( x ) d λ ( y ) + g ( a 1 , , a n , b 1 , , b n ) ,

with g ( a 1 , , a n , b 1 , , b n ) 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 ) :

Corollary 3.2

Let C = ( a i , b i , C i ) i I be an ordinal sum with I { 1 , , n } for some n N . Then

ξ ( C ) = i = 1 n ( b i a i ) 2 ξ ( C i ) + 6 f ( a 1 , , a n , b 1 , , b n )

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 .

Example 3.3

Consider C s = ( a i , b i , C i ) i { 1 , 2 } , whereby a 1 0 , a 2 s , b 1 s , and b 2 1 for s [ 0 , 1 ] and set C 1 = Π as well as C 2 = M . Figure 1 depicts the support of μ C s for different choices of s [ 0 , 1 ] . Using Corollary 3.2 we have ξ ( C s ) = ( 1 s ) 2 + 6 s 2 3 + s ( 1 s ) + ( 1 s ) 2 3 1 3 = 1 s 2 . Therefore, the map φ : [ 0 , 1 ] [ 0 , 1 ] defined by s ξ ( C s ) is continuous and onto. The same holds for ζ 1 ( C s ) = 1 s 3 .

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

(7) μ A = 1 3 μ A 1 f 12 + 1 3 μ A 2 f 23 + 1 3 μ A 3 f 31 ,

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 ).

Theorem 3.4

If A C has maximal d -asymmetry, i.e., if δ ( A ) = 3 d ( A , A t ) = 1 holds, then ξ satisfies ξ ( A ) 2 3 , 1 . Moreover, for every s 2 3 , 1 there exists a copula A with δ ( A ) = 1 fulfilling ξ ( A ) = s .

Proof

We may assume that A 1 3 , 2 3 = 1 3 and A 2 3 , 1 3 = 0 . Then there exist copulas A 1 , A 2 , A 3 C such that μ A = 1 3 μ A 2 f 12 + 1 3 μ A 3 f 23 + 1 3 μ A 1 f 31 holds. Defining h : [ 0 , 1 ] [ 0 , 1 ] by

h ( x ) = 1 3 + x if x 0 , 2 3 x 2 3 if x 2 3 , 1 ,

we have h T b and S h ( A ) = i 1 3 , i 3 , A i i { 1 , 2 , 3 } . Applying Lemmas 2.1 and 3.2 we therefore obtain

ξ ( A ) = ξ ( S h ( A ) ) = 6 f ( a 1 , , a n , b 1 , , b n ) + i = 1 3 1 9 ξ ( A i ) = 2 3 + i = 1 3 1 9 ξ ( A i ) 2 3 ,

with equality if and only if A i = Π for every i = 1 , 2 , 3 .

Defining A s by μ A s 1 3 μ C s f 12 + 1 3 μ C s f 23 + 1 3 μ C s f 31 with C s as in Example 3.3 yields

ξ ( A s ) = ξ ( S h ( A s ) ) = 1 3 ξ ( C s ) + 2 3 .

Considering ξ ( C s ) = 1 s 2 [ 0 , 1 ] for s [ 0 , 1 ] and using the same arguments as in Example 3.3 it follows that for every s 0 2 3 , 1 we find a copula A C with ξ ( A ) = s 0 and 3 d ( A , A t ) = δ ( A ) = 1 .□

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.

Theorem 3.5

If A C has maximal d -asymmetry, then ζ 1 satisfies ζ 1 ( A ) 5 6 , 1 . Furthermore, for every s 5 6 , 1 there exists a copula A with δ ( A ) = 1 fulfilling ζ 1 ( A ) = s .

Proof

Proceeding as in the proof of Theorem 3.4 we obtain S h ( A ) = i 1 3 , i 3 , A i i { 1 , 2 , 3 } . Considering the (SI)-rearrangement S h ( A ) of S h ( A ) it is clear that S h ( A ) is an ordinal sum again and can be expressed as S h ( A ) = i 1 3 , i 3 , A i i { 1 , 2 , 3 } . Since every A i is SI and hence fulfills A i ( x , y ) Π ( x , y ) = Π ( x , y ) for every ( x , y ) [ 0 , 1 ] 2 and every i { 1 , 2 , 3 } (see, e.g., [22][Section 5.2]), we obtain that

S h ( A ) C Π i 1 3 , i 3 , Π i { 1 , 2 , 3 }

holds pointwise. Due to the fact that ζ 1 is monotone w.r.t. the pointwise order on C and ζ 1 is invariant w.r.t. to (SI)-rearrangements (see [26]), we obtain

ζ 1 ( A ) = ζ 1 ( S h ( A ) ) = ζ 1 ( S h ( A ) ) ζ 1 ( C Π ) = 5 6 ,

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 3.3 (see Appendix A).□

While the minimum value of ξ for a copula A C with maximal d -asymmetry is attained if and only if A i = Π for every i = 1 , 2 , 3 in equation (7), ζ 1 exhibits a different behavior as demonstrated in the following example:

Example 3.6

Let A 1 C be defined by

A 1 ( x , y ) = x y + 1 2 x ( 1 x ) y ( 1 y ) .

Then a version of the corresponding Markov kernel of A 1 is given by K A 1 ( x , [ 0 , y ] ) = y + 1 2 ( 2 x 1 ) y ( y 1 ) . Furthermore, we set A 3 = A 1 and A 2 = Π and let A denote the ordinal sum given by A i 1 3 , i 3 , A i i { 1 , 2 , 3 } and C Π be the ordinal sum given by C Π i 1 3 , i 3 , Π i { 1 , 2 , 3 } (Figure 2).

By construction we have A C Π , however, considering

[ 0 , 1 ] [ 0 , 1 ] 1 2 ( 2 x 1 ) y ( y 1 ) d λ ( x ) d λ ( y ) = 0

yields

[ 0 , 1 ] [ 0 , 1 ] K Π ( x , [ 0 , y ] ) y 3 d λ ( x ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] 2 y 3 d λ ( x ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] 2 y 3 + 1 2 ( 2 x 1 ) y ( y 1 ) d λ ( x ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] 2 y 3 + 1 2 ( 2 x 1 ) y ( y 1 ) d λ ( x ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] K A 1 ( x , [ 0 , y ] ) y 3 d λ ( x ) d λ ( y )

and analogously we obtain

[ 0 , 1 ] [ 0 , 1 ] K Π ( x , [ 0 , y ] ) 2 3 + y 3 d λ ( x ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] K A 1 ( x , [ 0 , y ] ) 2 3 + y 3 d λ ( x ) d λ ( y ) .

Applying Lemma 3.1 we obtain ζ 1 ( A ) = ζ 1 ( C Π ) = 5 6 .

Figure 2 
               Density of the SI copulas 
                     
                        
                        
                           
                              
                                 C
                              
                              
                                 Π
                              
                           
                        
                        {C}_{\Pi }
                     
                   (left panel) and 
                     
                        
                        
                           A
                        
                        A
                     
                   (right panel) considered in Example 3.6. Although 
                     
                        
                        
                           A
                           
                              (
                              
                                 x
                                 ,
                                 y
                              
                              )
                           
                           ≥
                           
                              
                                 C
                              
                              
                                 Π
                              
                           
                           
                              (
                              
                                 x
                                 ,
                                 y
                              
                              )
                           
                        
                        A\left(x,y)\ge {C}_{\Pi }\left(x,y)
                     
                   holds for every 
                     
                        
                        
                           
                              (
                              
                                 x
                                 ,
                                 y
                              
                              )
                           
                           ∈
                           
                              
                                 
                                    [
                                    
                                       0
                                       ,
                                       1
                                    
                                    ]
                                 
                              
                              
                                 2
                              
                           
                        
                        \left(x,y)\in {\left[0,1]}^{2}
                     
                   and there exists some 
                     
                        
                        
                           
                              (
                              
                                 x
                                 ,
                                 y
                              
                              )
                           
                        
                        \left(x,y)
                     
                   with 
                     
                        
                        
                           A
                           
                              (
                              
                                 x
                                 ,
                                 y
                              
                              )
                           
                           >
                           
                              
                                 C
                              
                              
                                 Π
                              
                           
                           
                              (
                              
                                 x
                                 ,
                                 y
                              
                              )
                           
                        
                        A\left(x,y)\gt {C}_{\Pi }\left(x,y)
                     
                   we have 
                     
                        
                        
                           
                              
                                 ζ
                              
                              
                                 1
                              
                           
                           
                              (
                              
                                 A
                              
                              )
                           
                           =
                           
                              
                                 ζ
                              
                              
                                 1
                              
                           
                           
                              (
                              
                                 
                                    
                                       C
                                    
                                    
                                       Π
                                    
                                 
                              
                              )
                           
                        
                        {\zeta }_{1}\left(A)={\zeta }_{1}\left({C}_{\Pi })
                     
                  . On the contrary, 
                     
                        
                        
                           ξ
                        
                        \xi 
                     
                   fulfills 
                     
                        
                        
                           ξ
                           
                              (
                              
                                 A
                              
                              )
                           
                           >
                           ξ
                           
                              (
                              
                                 
                                    
                                       C
                                    
                                    
                                       Π
                                    
                                 
                              
                              )
                           
                        
                        \xi \left(A)\gt \xi \left({C}_{\Pi })
                     
                  .
Figure 2

Density of the SI copulas C Π (left panel) and A (right panel) considered in Example 3.6. Although A ( x , y ) C Π ( x , y ) holds for every ( x , y ) [ 0 , 1 ] 2 and there exists some ( x , y ) with A ( x , y ) > C Π ( x , y ) we have ζ 1 ( A ) = ζ 1 ( C Π ) . On the contrary, ξ fulfills ξ ( A ) > ξ ( C Π ) .

Remark 3.7

Considering the monotonicity of ζ 1 with respect to the pointwise order on C as proved in [26], Example 3.6 shows that there exist copulas A , B C with A B pointwise and A ( x , y ) < B ( x , y ) for some ( x , y ) [ 0 , 1 ] 2 fulfilling ζ 1 ( A ) = ζ 1 ( B ) .

4 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

C κ = 1 { A C : κ ( A ) 2 D 1 ( A , A t ) = 1 } C ,

the subclass of mutually completely dependent copulas is denoted by C m c d κ = 1 . We start with the family of mutually completely dependent copulas and show closedness w.r.t. the metric D .

Proposition 4.1

The set C m c d κ = 1 is closed in ( C , D ) .

Proof

Let ( A h n ) 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 C c d and A t C c d . Using [27, Lemma 10] there exist λ -preserving transformations g , g T such that a version of the Markov kernel K A ( , ) and K A t ( , ) is given by K A ( x , E ) = 1 E ( g ( x ) ) and K A t ( x , E ) = 1 E ( g ( x ) ) , respectively. Furthermore, since a copula A is completely dependent if and only if it is left-invertible w.r.t. the -product (see [27]) we have M = A t A . Applying Lemma 2.2 therefore yields that g g ( x ) = i d ( x ) for λ -a.e. x [ 0 , 1 ] . Using the fact that g is surjective λ -almost everywhere, there exists a λ -preserving and bijective transformation h T b such that h = g holds λ -a.e., implying A = A h C m c d . It remains to show that D 1 ( A h , A h t ) = 1 2 , which can be done as follows. Using [13][Theorem 3.5] and the triangle inequality we obtain

1 2 = [ 0 , 1 ] h n h n 1 d λ ( x ) [ 0 , 1 ] h n h d λ ( x ) + [ 0 , 1 ] h h 1 d λ ( x ) + [ 0 , 1 ] h 1 h n 1 d λ ( x )

for every n N . Applying [27][Proposition 15 (ii)] yields

D 1 ( A h , A h t ) = [ 0 , 1 ] h ( x ) h 1 ( x ) d λ ( x ) 1 2 [ D 1 ( A h n , A h ) + D 1 ( A h n t , A h t ) ] = 1 2 D ( A h n , A h ) .

Together with the fact that the maximal distance cannot exceed 1 2 it follows that D 1 ( A h , A h t ) = 1 2 , which completes the proof.□

The following example shows that the set C m c d κ = 1 is not closed w.r.t. the metric D 1 .

Example 4.2

Let A h n C m c d be the mutually completely dependent copula induced by the bijective measure-preserving transformation h n : [ 0 , 1 ] [ 0 , 1 ] , given by

h n ( x ) = x + j 1 n if x j 1 n , j n and j 1 , , n 2 x 1 + j n if x j 1 n , j n and j n 2 + 1 , , n 1 if x = 1

for all n 2 N and let A h C c d be the completely dependent copula induced by the λ -preserving transformation h : [ 0 , 1 ] [ 0 , 1 ] given by h ( x ) 2 x ( m o d 1 ) (see Figure 1 in [9]). Setting C n i 1 4 , i 4 , A h 2 n i { 1 , 2 , 3 , 4 } and C i 1 4 , i 4 , A h i { 1 , 2 , 3 , 4 } we have C n C m c d and C C c d and according to [9, Example 3.3] it is straightforward to verify that lim n D 1 ( C n , C ) = 0 . As a next step, we reorder the shrunk copulas to obtain maximal D 1 -asymmetry. Let f denote the λ -preserving interval exchange transformation f : [ 0 , 1 ] [ 0 , 1 ] defined by f ( x ) ( x 1 4 ) 1 1 4 , 1 ( x ) + ( x + 3 4 ) 1 0 , 1 4 ( x ) and, furthermore, let S f ( C n ) C m c d and S f ( C ) C c d denote the respective shuffles (Figure 3). Due to the fact that the metric D 1 is shuffle-invariant w.r.t. bijective transformations (using the same arguments as in the proof of Lemma 2.1) yields

lim n D 1 ( S f ( C n ) , S f ( C ) ) = lim n D 1 ( C n , C ) = 0 .

Setting U 0 , 1 4 3 4 , 1 and considering property (3) of Theorem 4.1 in [13] (see also Theorem 4.4 (iii) in the sequel) we directly obtain that S f ( C n ) and S f ( C ) are maximal asymmetric w.r.t. D 1 , which shows that C m c d κ = 1 is not closed w.r.t. the metric D 1 .

Figure 3 
               The support of the copulas 
                     
                        
                        
                           
                              
                                 μ
                              
                              
                                 
                                    
                                       S
                                    
                                    
                                       f
                                    
                                 
                                 
                                    (
                                    
                                       
                                          
                                             C
                                          
                                          
                                             n
                                          
                                       
                                    
                                    )
                                 
                              
                           
                        
                        {\mu }_{{{\mathcal{S}}}_{f}\left({C}_{n})}
                     
                   (black) for 
                     
                        
                        
                           n
                           =
                           4
                        
                        n=4
                     
                   (left panel) and 
                     
                        
                        
                           n
                           =
                           8
                        
                        n=8
                     
                   (right panel) as well as the copula 
                     
                        
                        
                           
                              
                                 μ
                              
                              
                                 
                                    
                                       S
                                    
                                    
                                       f
                                    
                                 
                                 
                                    (
                                    
                                       C
                                    
                                    )
                                 
                              
                           
                        
                        {\mu }_{{{\mathcal{S}}}_{f}\left(C)}
                     
                   (magenta) as considered in Example 4.2.
Figure 3

The support of the copulas μ S f ( C n ) (black) for n = 4 (left panel) and n = 8 (right panel) as well as the copula μ S f ( C ) (magenta) as considered in Example 4.2.

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].

Lemma 4.3

For every A , B , C C , the following inequality holds for every p [ 1 , ) :

D p p ( A B , A C ) D p p ( B , C ) .

Proof

Applying Lemma 2.2, Jensen’s inequality, disintegration and using the fact that μ A is doubly stochastic we obtain

D p p ( A B , A C ) = [ 0 , 1 ] [ 0 , 1 ] [ 0 , 1 ] K B ( t , [ 0 , y ] ) K A ( x , d t ) [ 0 , 1 ] K C ( t , [ 0 , y ] ) K A ( x , d t ) p d λ ( x ) d λ ( y ) [ 0 , 1 ] [ 0 , 1 ] [ 0 , 1 ] K B ( t , [ 0 , y ] ) K C ( t , [ 0 , y ] ) p K A ( x , d t ) d λ ( x ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] 2 K B ( t , [ 0 , y ] ) K C ( t , [ 0 , y ] ) p d μ A ( x , t ) d λ ( y ) = [ 0 , 1 ] [ 0 , 1 ] K B ( t , [ 0 , y ] ) K C ( t , [ 0 , y ] ) p d λ ( t ) d λ ( y ) = D p p ( B , C ) ,

which completes the proof.□

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 C the following statements are equivalent:

  1. κ ( A ) = 1 ,

  2. Φ A , A t ( 1 2 ) = 1 (or equivalently, A has maximal D -asymmetry),

  3. there exists a Borel set U ( [ 0 , 1 ] ) with the following properties:

    λ U 0 , 1 2 = λ U 1 2 , 1 = 1 4 , μ A U × 0 , 1 2 = 1 2 , μ A 0 , 1 2 × U = 0 ,

  4. there exist sets U 1 , U 2 ( [ 0 , 1 ] ) with U 1 0 , 1 2 , U 2 1 2 , 1 , λ ( U 1 ) = λ ( U 2 ) = 1 4 and V 1 0 , 1 2 U 1 and V 2 1 2 , 1 U 2 , and copulas C 1 , C 2 , C 3 , C 4 C such that the following identity

    A ( x , y ) = 1