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BY 4.0 license Open Access Published by De Gruyter Open Access November 10, 2023

A link between Kendall’s τ, the length measure and the surface of bivariate copulas, and a consequence to copulas with self-similar support

  • Juan Fernández Sánchez and Wolfgang Trutschnig EMAIL logo
From the journal Dependence Modeling

Abstract

Working with shuffles, we establish a close link between Kendall’s τ , the so-called length measure, and the surface area of bivariate copulas and derive some consequences. While it is well known that Spearman’s ρ of a bivariate copula A is a rescaled version of the volume of the area under the graph of A , in this contribution we show that the other famous concordance measure, Kendall’s τ , allows for a simple geometric interpretation as well – it is inextricably linked to the surface area of A .

1 Introduction

Spearman’s ρ of a bivariate copula A is a rescaled version of the volume of the area below the graph of A (see [3,14]) in the sense that

ρ ( A ) = 12 [ 0 , 1 ] 2 A d λ 2 3

holds. Letting [ A ] t { ( x , y ) [ 0 , 1 ] 2 : A ( x , y ) t } denote the lower t -cut of A for every t [ 0 , 1 ] , and applying Fubini’s theorem directly yields

ρ ( A ) = 12 [ 0 , 1 ] λ 2 ( [ A ] t ) d λ ( t ) 3 ,

which lead Coblenz et al. [1] to conjecture that adequately rescaling the so-called length measure ( A ) of A , defined as the average arc-length of the contour lines of A , might result in a (new or already known) concordance measure. The conjecture was falsified in [1], and only some but not all properties of a concordance measure are fulfilled, in particular, we do not have continuity with respect to pointwise convergence of copulas in general.

Motivated by the aforementioned facts, the objective of this note is twofold: we first derive the somewhat surprising result that on a subfamily of bivariate copulas – the class C mcd of all mutually completely dependent copulas (including all classical shuffles), which is dense in the class C of all bivariate copulas with respect to uniform convergence – the length measure is, in fact, an affine transformation of Kendall’s τ , and vice versa. As a consequence, the length measure restricted to C mcd is continuous with respect to pointwise convergence of copulas. We then focus on the surface area of bivariate copulas and derive analogous statements, i.e., on the class C mcd , the surface area is an affine transformation of Kendall’s τ (and hence of the length measure) too. For obtaining both main results, the key will be a simple geometric identity linking the length of measure and the surface area with the area of the set Ω 2 , which is given as follows:

(1) Ω 2 A h = { ( x , y ) [ 0 , 1 ] 2 : h ( x ) y , h 1 ( y ) x } ,

where h denotes the transformation corresponding to the completely dependent copula A h .

The remainder of this note is organized as follows: in Section 2, we gather preliminaries and notations that will be used in the sequel; Section 3 is the core of the article, derives the aforementioned identities linking Kendall’s τ , the length of measure, and the surface of mutually completely dependent copulas, provides a geometrical interpretation, and shows that outside the class C mcd , these identities need not hold; as a small (but mathematically interesting) application of the established relationships, Section 4 derives simple formulas for the length of measure, and the surface area of completely dependent copulas with self-similar support, which without the aforementioned identities, seem very hard to establish. Finally, Section 5 provides an outlook on related questions to be tackled in the future.

2 Notations and preliminaries

In the sequel, we will let C denote the family of all bivariate copulas. For each copula C C , the corresponding doubly stochastic measure will be denoted by μ C , i.e., μ C ( [ 0 , x ] × [ 0 , y ] ) = C ( x , y ) holds for all x , y [ 0 , 1 ] . Considering the uniform metric d on C , it is well known that ( C , d ) is a compact metric space and that in C , pointwise and uniform convergence are equivalent. For more background on copulas and doubly stochastic measures, we refer to [3,14].

For every metric space, ( Ω , d ) the Borel σ -field in Ω will be denoted by ( Ω ) . The Lebesgue measure on the Borel σ -field ( [ 0 , 1 ] 2 ) of [ 0 , 1 ] 2 will be denoted by λ 2 , the univariate version on ( [ 0 , 1 ] ) by λ . Given probability spaces ( Ω , A , P ) and ( Ω , A , P ) and a measurable transformation T : Ω Ω , the push forward of P via T will be denoted by P T , i.e., P T ( F ) = P ( T 1 ( F ) ) for all F A .

In what follows, Markov kernels will be a handy tool. 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 if 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 almost surely (a.s.). It is well known that a regular conditional distribution of Y given X exists and is unique P X a.s. 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 ( , ) and directly be interpreted as mapping from K A : [ 0 , 1 ] × ( [ 0 , 1 ] ) [ 0 , 1 ] . Note that for every A C and Borel sets E , F ( [ 0 , 1 ] ) , we have the following disintegration formulas:

(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 [10,12].

A copula A C will be called completely dependent (or functionally dependent) if there exists some λ -preserving transformation h : [ 0 , 1 ] [ 0 , 1 ] (i.e., a transformation with λ h = λ ) such that K ( x , E ) = 1 E ( h ( x ) ) is a Markov kernel of A . The copula induced by h will be denoted by A h , and the class of all completely dependent copulas is denoted by C c d . A completely dependent copula A h is called mutually completely dependent if the transformation h is bijective. Note that, in this case, the transpose A h t of A h , defined by A h t ( x , y ) = A h ( y , x ) , coincides with A h 1 . The family of all mutually completely dependent copulas will be denoted by C mcd . Note that mutually completely dependent copulas model the seemingly pathological case of pairs ( X , Y ) of uniform [ 0 , 1 ] random variables X and Y such that Y is a measurable function of X , and vice versa. It is well known (see [3,14]), however, that C mcd is dense in ( C , d ) , and in fact, even the family of all equidistant even shuffles (again see [3,14]) is dense. This very observation led to the observation that d is not able to discriminate between independence and complete dependence, which, in turn, triggered the study of stronger metrics overcoming that problem by Trutschnig [16]. For further properties of completely dependent copulas, we refer to [16] and the references therein.

Turning toward the length profile introduced and studied by Coblenz et al. [1], let Γ A , t denote the boundary of the lower t -cut [ A ] t in ( 0 , 1 ) 2 and H 1 ( Γ A , t ) it is arc-length. Then, the length profile of A is defined as the function L A : [ 0 , 1 ] [ 0 , ) , given by

L A ( t ) = H 1 ( Γ A , t ) .

It is easy to verify that

(3) 2 ( 1 t ) L A ( t ) 2 ( 1 t )

holds for every t ( 0 , 1 ) . Building upon L A , the so-called length measure ( A ) of A is defined as

(4) ( A ) = ( 0 , 1 ) L A ( t ) d λ ( t )

and describes the average arc-length of upper t -cuts of A . Using this in equation (3) immediately yields

(5) ( W ) = 1 2 ( A ) 1 = ( M ) ,

as well as ( A ) [ 1 2 , 1 ] (equation (5) was also one of the reasons for falsely conjecturing that the length measure might be transformable into a concordance measure).

Coblenz et al. [1] showed that for mutually completely dependent copulas A h , the length profile allows for a simple calculation. In fact, using the co-area formula, we have

( A h ) = ( 0 , 1 ) 2 A h ( u , v ) 2 d λ 2 ( u , v ) ,

where A h denotes the gradient of A h (whose existence λ 2 -almost everywhere is assured by Rademacher’s theorem and Lipschitz continuity, see [5]). The last equation simplifies to the nice identity

(6) ( A h ) = 1 ( 2 2 ) λ 2 ( Ω 2 ) ,

with

(7) Ω 2 A h Ω 2 = { ( u , v ) ( 0 , 1 ) 2 : A h ( u , v ) 2 = 2 } = { ( u , v ) ( 0 , 1 ) 2 : h ( u ) v , h 1 ( v ) u } .

According to Coblenz et al. [1] (also see the proof of Lemma 3.4) the λ 2 -measure of the set Ω 2 coincides with the one of the set Ω 0 , defined as follows:

(8) Ω 0 A h Ω 0 = { ( u , v ) ( 0 , 1 ) 2 : A h ( u , v ) 2 = 0 } = { ( u , v ) ( 0 , 1 ) 2 : h ( u ) > v , h 1 ( v ) > u } .

Obviously Ω 2 is the set of all points ( u , v ) [ 0 , 1 ] 2 “above,” and Ω 0 the set of all points ( u , v ) [ 0 , 1 ] 2 “below” the graphs of h and h 1 . Note that for classical equidistant straight shuffles, equation (6) implies that ( A h ) can be calculated by simply counting squares as Figure 1 illustrates in terms of two simple examples – one shuffle with three, and a second one with nice equidistant stripes. Throughout the rest of this note, we will only write Ω 2 instead of Ω 2 A h and Ω 0 instead of Ω 0 A h whenever no confusion will arise.

Figure 1 
               The sets 
                     
                        
                        
                           
                              
                                 Ω
                              
                              
                                 0
                              
                           
                        
                        {\Omega }_{0}
                     
                   (in magenta) and 
                     
                        
                        
                           
                              
                                 Ω
                              
                              
                                 
                                    
                                       2
                                    
                                 
                              
                           
                        
                        {\Omega }_{\sqrt{2}}
                     
                   (in green) for an even shuffles of three strips (left panel) and nine strips (right panel). In this case, we have 
                     
                        
                        
                           
                              
                                 λ
                              
                              
                                 2
                              
                           
                           
                              (
                              
                                 
                                    
                                       Ω
                                    
                                    
                                       
                                          
                                             2
                                          
                                       
                                    
                                 
                              
                              )
                           
                           =
                           
                              
                                 1
                              
                              
                                 9
                              
                           
                        
                        {\lambda }_{2}\left({\Omega }_{\sqrt{2}})=\frac{1}{9}
                     
                   and 
                     
                        
                        
                           ℓ
                           
                              (
                              
                                 
                                    
                                       A
                                    
                                    
                                       h
                                    
                                 
                              
                              )
                           
                           =
                           1
                           
                           ‒
                           
                           
                              (
                              
                                 2
                                 
                                 ‒
                                 
                                 
                                    
                                       2
                                    
                                 
                              
                              )
                           
                           
                              
                                 1
                              
                              
                                 9
                              
                           
                        
                        \ell \left({A}_{h})=1\hspace{0.33em}‒\hspace{0.33em}\left(2\hspace{0.33em}‒\hspace{0.33em}\sqrt{2})\frac{1}{9}
                     
                   for the first shuffle and 
                     
                        
                        
                           
                              
                                 λ
                              
                              
                                 2
                              
                           
                           
                              (
                              
                                 
                                    
                                       Ω
                                    
                                    
                                       
                                          
                                             2
                                          
                                       
                                    
                                 
                              
                              )
                           
                           =
                           
                              
                                 1
                              
                              
                                 9
                              
                           
                           +
                           
                              
                                 1
                              
                              
                                 27
                              
                           
                        
                        {\lambda }_{2}\left({\Omega }_{\sqrt{2}})=\frac{1}{9}+\frac{1}{27}
                     
                   and 
                     
                        
                        
                           ℓ
                           
                              (
                              
                                 
                                    
                                       A
                                    
                                    
                                       h
                                    
                                 
                              
                              )
                           
                           =
                           1
                           
                           ‒
                           
                           
                              (
                              
                                 2
                                 
                                 ‒
                                 
                                 
                                    
                                       2
                                    
                                 
                              
                              )
                           
                           
                              
                                 
                                    
                                       1
                                    
                                    
                                       9
                                    
                                 
                                 +
                                 
                                    
                                       1
                                    
                                    
                                       27
                                    
                                 
                              
                           
                        
                        \ell \left({A}_{h})=1\hspace{0.33em}‒\hspace{0.33em}\left(2\hspace{0.33em}‒\hspace{0.33em}\sqrt{2})\left(\frac{1}{9}+\frac{1}{27}\right)
                     
                   for the second one.
Figure 1

The sets Ω 0 (in magenta) and Ω 2 (in green) for an even shuffles of three strips (left panel) and nine strips (right panel). In this case, we have λ 2 ( Ω 2 ) = 1 9 and ( A h ) = 1 ( 2 2 ) 1 9 for the first shuffle and λ 2 ( Ω 2 ) = 1 9 + 1 27 and ( A h ) = 1 ( 2 2 ) 1 9 + 1 27 for the second one.

3 The interrelations

We now derive a simple formula linking Kendall’s τ and the length measure for mutually completely dependent copula and start with some preliminary observations. Working with checkerboard copulas, using integration by parts (see [14]), and finally applying an approximation result like Theorem 3.2 in the study by Kasper et al. [11] yields that for arbitrary bivariate copulas A , B C the following identity holds:

(9) τ ( A ) = 4 [ 0 , 1 ] 2 A d μ A 1 = 4 1 2 [ 0 , 1 ] 2 K A ( x , [ 0 , y ] ) K A t ( y , [ 0 , x ] ) d λ 2 ( x , y ) 1 .

For A h C mcd , equation (9) can be derived in the following simple alternative way, which we include for the sake of completeness: Using the fact that for A h C mcd and every x [ 0 , 1 ] , we have

A ( x , h ( x ) ) = [ 0 , x ] K A h ( t , [ 0 , h ( x ) ] ) d λ ( t ) = [ 0 , x ] K A h ( t , [ 0 , h ( x ) ) ) d λ ( t ) = [ 0 , x ] 1 [ 0 , h ( x ) ) ( h ( t ) ) d λ ( t ) = [ 0 , x ] 1 1 [ h ( x ) , 1 ] ( h ( t ) ) d λ ( t ) = x [ 0 , 1 ] 1 [ 0 , x ] ( t ) 1 [ h ( x ) , 1 ] ( h ( t ) ) d λ ( t ) .

Using disintegration and change of coordinates directly yields

[ 0 , 1 ] 2 A d μ A = [ 0 , 1 ] A ( x , h ( x ) ) d λ ( x ) = 1 2 [ 0 , 1 ] [ 0 , 1 ] 1 [ 0 , x ] ( t ) 1 [ h ( x ) , 1 ] ( h ( t ) ) d λ ( t ) d λ ( x )

and hence proves equation (9). The latter identity, however, boils down to an affine transformation of λ 2 ( Ω 2 ) by considering

[ 0 , 1 ] 2 A d μ A = 1 2 [ 0 , 1 ] [ 0 , 1 ] 1 [ 0 , x ] ( h 1 h ( t ) ) 1 [ h ( x ) , 1 ] ( h ( t ) ) d λ ( t ) d λ ( x ) = 1 2 [ 0 , 1 ] [ 0 , 1 ] 1 [ 0 , x ] ( h 1 ( y ) ) 1 [ h ( x ) , 1 ] ( y ) d λ ( y ) d λ ( x ) = 1 2 λ 2 ( Ω 2 ) .

Having this, the identity

(10) τ ( A h ) = 4 1 2 λ 2 ( Ω 2 ) 1 = 1 4 λ 2 ( Ω 2 ) ,

follows immediately. Note that equation (10) implies that the area of Ω 2 coincides with the quantity inv(h) as studied in [15, Lemma 3.1]. Comparing equations (6) and (10) shows the existence of an affine transformation a : [ 1 , 1 ] [ 1 2 , 1 ] such that

a ( τ ( A h ) ) = ( A h )

holds for every A h C mcd ; in other words, we have proved the subsequent result:

Theorem 3.1

For every A h C mcd , the following identity linking the length of measure and Kendall’s τ holds:

(11) ( A h ) = 1 2 2 4 ( 1 τ ( A h ) ) .

Theorem 3.1 provides a positive answer to the question posed in [1], “whether there are links between the length of level curves and concordance measures” – even the conjectured “weighting” mentioned in [1] is not necessary, in the class C mcd all we need is a fixed affine transformation.

In the study by Coblenz et al. [1], it was further shown that the length measure interpreted as function : C [ 2 2 , 1 ] is not continuous w.r.t. d . The previous result implies, however, that within the dense subclass C mcd , the length measure is indeed continuous.

Corollary 3.2

The mapping : C mcd [ 1 2 , 1 ] is continuous with respect to d .

Proof

Suppose that A h , A h 1 , A h 2 , are mutually completely dependent copulas and that the sequence ( A h n ) n N converges to A h pointwise. Being a concordance measure, Kendall’s τ is continuous with respect to d , so we have lim n τ ( A h n ) = τ ( A h ) and equation (10) directly yields lim n ( A h n ) = ( A h ) .□

Corollary 3.3

For every z [ 1 2 , 1 ] , there exists some mutually completely dependent copula A h with ( A h ) = z . In other words, all values in [ 1 2 , 1 ] are attained by .

Proof

According to Schreyer et al. [15], for each ( x , y ) in the region determined by Kendall’ τ and Spearman’s ρ , there exists some mutually completely dependent copula C h fulfilling

( τ ( A h ) , ρ ( A h ) ) = ( x , y ) .

Having this, the result directly follows via equation (10).□

Moving away from the length measure, we now turn to the surface area of copulas, derive analogous statements, and start with showing yet another simple formula for elements in C mcd . Considering that copulas are Lipschitz continuous, the surface area surf ( A ) of an arbitrary copula A is given as follows:

(12) surf ( A ) = [ 0 , 1 ] 2 A x ( x , y ) 2 + A y ( x , y ) 2 + 1 d λ 2 ( x , y ) = [ 0 , 1 ] 2 K A ( x , [ 0 , y ] ) 2 + K A t ( y , [ 0 , x ] ) 2 + 1 d λ 2 ( x , y ) .

Again working with mutually completely dependent copulas yields the following result:

Lemma 3.4

For every A h C mcd , the surface area of A h is given as follows:

(13) surf ( A h ) = 2 ( 2 2 1 3 ) λ 2 ( Ω 2 ) .

Proof

For the case of a completely dependent copula A h equation (12) obviously simplifies to

surf ( A h ) = [ 0 , 1 ] 2 1 [ 0 , y ] 2 ( h ( x ) ) + 1 [ 0 , x ] 2 ( h 1 ( y ) ) + 1 d λ 2 ( x , y ) = [ 0 , 1 ] 2 1 [ 0 , y ] ( h ( x ) ) + 1 [ 0 , x ] ( h 1 ( y ) ) + 1 d λ 2 ( x , y ) .

Considering that the latter integrand is a step function only attaining the values 1 , 2 and 3 , defining

Ω 0 A h Ω 0 = { ( x , y ) ( 0 , 1 ) 2 : h ( x ) > y , h 1 ( y ) > x }

and ( Ω 2 as before)

Ω 1 A h Ω 1 = [ 0 , 1 ] 2 \ ( Ω 2 Ω 0 ) ,

we therefore have

surf ( A h ) = λ 2 ( Ω 0 ) + 2 λ 2 ( Ω 1 ) + 3 λ 2 ( Ω 2 ) .

The latter identity can be further simplified: the measurable bijection Ψ h : [ 0 , 1 ] 2 [ 0 , 1 ] 2 , defined by Ψ h ( x , y ) = ( h 1 ( y ) , h ( x ) ) , obviously fulfills λ 2 Ψ h = λ 2 . Therefore, using the fact that

Ψ h 1 ( Ω 0 ) = { ( x , y ) [ 0 , 1 ] 2 : ( h 1 ( y ) , h ( x ) ) Ω 0 } = { ( x , y ) [ 0 , 1 ] 2 : h ( x ) y , h 1 ( y ) x } = Ω 2 ,

it follows that λ 2 ( Ω 0 ) = λ 2 ( Ω 2 ) holds. This altogether yields

surf ( A h ) = λ 2 ( Ω 2 ) + 2 λ 2 ( Ω 1 ) + 3 λ 2 ( Ω 2 ) = ( 1 + 3 ) λ 2 ( Ω 2 ) + 2 ( 1 2 λ 2 ( Ω 2 ) ) = 2 + ( 1 + 3 2 2 ) < 0 λ 2 ( Ω 2 ) ,

which completes the proof.□

Theorem 3.5

For every A h C mcd , the following identity linking the surface area and Kendall’s τ holds:

(14) surf ( A h ) = 2 2 2 1 3 4 ( 1 τ ( A h ) )

As in the case of the length measure, we have the following two immediate corollaries:

Corollary 3.6

The mapping surf : C mcd [ 1 + 3 2 , 2 ] is continuous with respect to d .

Corollary 3.7

For every z [ 1 + 3 2 , 2 ] , there exists some mutually completely dependent copula A h with surf ( A h ) = z . In other words, all values in [ 1 + 3 2 , 2 ] are attained by surf .

Combining Theorems 3.1 and 3.5 yields the following identity linking the length measure and the surface area of mutually completely dependent copulas.

Corollary 3.8

For every A h C mcd , the following identity holds:

(15) surf ( A h ) = 2 2 2 1 3 2 2 ( 1 ( A h ) ) .

Remark 3.9

The aforementioned interrelations lead to the following seemingly new interpretation of the interplay between the two most well known measures of concordance, Kendall’s τ and Spearman’s ρ , as studied in [2,4,15] (and the references therein): within the dense class C mcd maximizing/minimizing Kendall’s τ for a given value of Spearman’s ρ is equivalent to maximizing/minimizing the surface area of copulas for a given value of the volume. Determining the exact τ - ρ region (for which according to [15] considering all shuffles is sufficient), one might therefore be reminded of the famous isoperimetric inequality bounding the surface area of a set by a function of the volume (see [6]).

Example 3.10

Considering that all results established in this section – in particular, Theorem 3.1 linking Kendall’ τ with the length profile and Theorem 3.5 interrelating τ with the surface – have only been stated and proved for mutually completely dependent copulas, the question naturally arises: can they be extended to classes larger than or different to C mcd ? Since one naturally might conjecture that equations (11) and (14) additionally hold (at least) for sufficiently smooth copulas, complementing the results in the study by Coblenz et al. [1], we now focus on the latter identity and show that it does not even hold for the product copula Π . First, note that because of τ ( Π ) = 0 , the right hand-side of equation (14) simplifies to

2 2 2 1 3 4 ( 1 τ ( Π ) ) = 2 2 + 1 + 3 4 1.3901 1 + 3 2 , 2 .

On the other hand, calculating surf ( Π ) according to equation (12) yields

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

The latter integral can be calculated analytically, and after some tedious but straightforward steps, one finally obtains

surf ( Π ) = 6 3 π 18 + arsinh 1 2 + 1 36 log ( 1351 + 780 3 ) 1.2808 < 1 + 3 2 1.3660 .

In other words, equation (14) does not hold for Π , and more surprisingly, there is no element of C mcd whose surface is at least close to surf ( Π ) – for every element A h of C mcd , the difference surf ( A h ) surf ( Π ) is even larger than the diameter of the full range of surf restricted to C mcd . Note that this observation does not contradict the fact that C mcd is dense in ( C , d ) , and consequently, for every copula A C , there exists some sequence ( A h n ) n N in C mcd with lim n d ( A h n , A ) = 0 and lim n τ ( A h n ) = τ ( A ) . In fact, pointwise convergence of copulas does not imply convergence of the sequence ( K A h n ( x , [ 0 , y ] ) ) n N to K A ( x , [ 0 , y ] ) for a sufficiently large set of ( x , y ) [ 0 , 1 ] 2 (see [16] for a related discussion), hence (as illustrated by Π above) the corresponding surface areas may be quite different.

Example 3.10 implies that the mapping A surf ( A ) assigning each copula its surface area is not continuous on the full domain C with respect to d ; according to Corollary 3.6, however, its restriction to C mcd is continuous. Intuitively, this is not too surprising since – contrary to the general case where the integrand K A ( x , [ 0 , y ] ) 2 + K A t ( y , [ 0 , x ] ) 2 + 1 may attain arbitrary values in [ 1 , 3 ] – in the mutually completely dependent setting, the kernels only attain the values 0 and 1, so the integrand is a step function only attaining the values 1 , 2 and 3 .

One natural question, therefore, would be whether there are stronger topologies O on C with respect to which the mapping A surf ( A ) is indeed continuous. And, in the positive case, if there are “nice” subclasses which are dense in ( C , O ) . We will provide a positive answer to both questions and start with recalling a stronger notion of convergence of copulas going back to the study by Kasper et al. [11] and the concept of checkerboard approximations.

Definition 3.11

[11] Suppose that A , A 1 , A 2 , are copulas and let K A , K A 1 , K A 2 , be (versions of) the corresponding Markov kernels. We will say that ( A n ) n N converges weakly conditional to A if and only if for λ -almost every x [ 0 , 1 ] we have that the sequence ( K A n ( x , ) ) n N of probability measures on ( [ 0 , 1 ] ) converges weakly to the probability measure K A ( x , ) .

Following the study by Li et al. [13], we can define checkerboard copulas as follows: Fix N N and define the squares R i j N for i , j { 1 , , N } by

R i j N = i 1 N , i N × j 1 N , j N .

Definition 3.12

[13,16] A copula A N C is called N-checkerboard copula, if A N is absolutely continuous and (a version of) its density k A N is constant on the interior of each square R i j N . We refer to N as the resolution of A N , denote the set of all N -checkerboard copulas by C N , and set C = N = 1 C N .

For A C and N N , the (absolutely continuous) copula C B N ( A ) C N defined by

(16) C B N ( A ) ( x , y ) 0 x 0 y N 2 i , j = 1 N μ A ( R i j N ) 1 R i j N ( s , t ) d λ ( t ) d λ ( s )

is called N-checkerboard approximation of A or simply N -checkerboard of A .

Having that, the following approximation result for the mapping A surf ( A ) , saying that checkerboard approximations also serve as surface approximations, can be proved.

Theorem 3.13

Let A C be an arbitrary copula and C B N ( A ) be the N-checkerboard approximation on A. Then, the following identity holds:

(17) lim N surf ( C B N ( A ) ) = surf ( A ) .

Proof

According to Corollary 3.2 in the study by Li et al. [13], the sequence ( C B N ( A ) ) N N converges weakly conditional to A for N , and the same is true for the sequence ( C B N ( A t ) ) N N and A t . Considering C B N ( A t ) = ( C B N ( A ) ) t , we can proceed as follows: let Λ ( [ 0 , 1 ] ) denote a set with λ ( Λ ) = 1 such that for every x Λ , the sequence ( K C B N ( A ) ( x , ) ) N N converges weakly to K A ( x , ) , and let x Λ be arbitrary but fixed. Then, we have

lim N K C B N ( A ) ( x , [ 0 , y ] ) = K A ( x , [ 0 , y ] )

for every continuity point y [ 0 , 1 ] of the function z K A ( x , [ 0 , z ] ) , so in particular for λ -almost every y [ 0 , 1 ] . Moreover, since (by disintegration) for λ -almost every y [ 0 , 1 ] , we have K A t ( y , { x } ) = 0 , it follows that for λ -almost every y [ 0 , 1 ] , the point x is a continuity point of the function z K A t ( y , [ 0 , z ] ) , which implies that for λ -almost every y [ 0 , 1 ] ,

lim N K C B N ( A ) t ( y , [ 0 , x ] ) = K A t ( y , [ 0 , x ] )

holds. Having this, using Fubini’s theorem and dominated convergence, we conclude that

lim N [ 0 , 1 ] K C B N ( A ) ( x , [ 0 , y ] ) 2 + K C B N ( A ) t ( y , [ 0 , x ] ) 2 + 1 d λ ( y ) = [ 0 , 1 ] K A ( x , [ 0 , y ] ) 2 + K A t ( y , [ 0 , x ] ) 2 + 1 d λ ( y ) .

Considering that x Λ was arbitrary and that λ ( Λ ) = 1 , again using dominated convergence completes the proof.□

Remark 3.14

Theorem 3.13 remains valid if instead of checkerboards we consider so-called checkmin copulas in which case shrunk versions of M instead of Π constitute the mass distributions in the little squares R i j N (again see [13]). Contrary to checkerboards, working with checkmin copulas, the integrand in equation (12) is again a step function (as it is the case for all shuffles). Checkmin copulas are closely linked to equidistant even shuffles (a.k.a. classical shuffles of M ) – in fact, using Birkhoff’s famous theorem (see [9]) it is straightforward to show that each checkmin copula with resolution N can be expressed as convex combination of finitely many equidistant even shuffles (with N stripes). The reason for considering checkerboards in Theorem 3.13 is that they are more commonly encountered than checkmins.

4 Calculating τ , , and surf for mutually completely dependent copulas with self-similar support

Calculating even standard characteristics like Kendall’s τ for copulas with fractal supports is a difficult endeavor. As a (playful) by-product of the identities established in Section 3, we now show how simple formulas for τ , , and surf of mutually completely dependent copulas with self-similar support can be derived. We first recall the notion of so-called transformation matrices and the construction of copulas with fractal/self-similar support, then use these tools to construct mutually completely dependent copulas with self-similar support, and finally derive simple expressions for Kendall’s τ and the length of measure of copulas of this type.

Definition 4.1

[8,16,17] A n × m -matrix T = ( t i j ) i = 1 n , j = 1 m is called transformation matrix if it fulfills the following four conditions: (i) max ( n , m ) 2 , (ii), all entries are nonnegative, (iii) i , j t i j = 1 , and (iv) no row or column has all entries 0.

In other words, a transformation matrix is a probability distribution τ on ( , 2 ) with = I 1 × I 2 , I 1 = { 1 , , n } , and I 2 = { 1 , , m } such that τ ( { i } × I 2 ) > 0 for every i I 1 and τ ( I 1 × { j } ) > 0 for every j I 2 .

Given a transformation matrix T , define the vectors ( a j ) j = 0 m , ( b i ) i = 0 n of cumulative column and row sums by

(18) a 0 = b 0 = 0 a j = j 0 j i = 1 n t i j j { 1 , , m } b i = i 0 i j = 1 m t i j i { 1 , , n } .

Considering that T is a transformation matrix, both ( a j ) j = 0 m and ( b i ) i = 0 n are strictly increasing. Consequently, R j i [ a j 1 , a j ] × [ b i 1 , b i ] are compact nonempty rectangles for every j { 1 , , m } and i { 1 , , n } . Defining the contraction w j i : [ 0 , 1 ] 2 R j i by

w j i ( x , y ) = ( a j 1 + x ( a j a j 1 ) , b i 1 + x ( b i b i 1 ) )

therefore yields the iterated function system with probabilities { [ 0 , 1 ] 2 , ( w j i ) j = 1 m , i = 1 n , ( t i j ) j = 1 m , i = 1 n } . The induced operator V T on P C given by

(19) V T ( μ ) j = 1 m i = 1 n t i j μ w j i

is easily verified to be well defined (i.e., it maps P C into itself, again see [8,16,17]) – in the sequel, we will also consider V T as a transformation mapping C into itself. According to the study by Trutschnig [16], for every transformation matrix T , there exists a unique copula A T * with V T ( A T * ) = A T * such that

(20) lim n D 1 ( V T n ( B ) , A T * ) = 0

holds for arbitrary B C (i.e., A T * is the unique, globally attractive fixed point of V T ).

Suppose now that 2 N N and let π be a permutation of { 1 , , N } . Then, the matrix T π = ( t i j ) i = 1 N , j = 1 N , defined by

t i , j = 1 N 1 { j } ( π ( i ) ) , i , j { 1 , , N }

is obviously a transformation matrix. To simplify notation, we will simply write V π V T π and A T π * = A T * in the sequel. Obviously, V π does not only map C to C but also C mcd to C mcd . Considering that (see [16]) C c d is closed in ( C , D 1 ) using equation (20), it follows immediately that A π * C mcd , so there exists some λ -preserving bijection h π * with A π * = A h * . Since the support of A π * is self-similar, it seems intractable to calculate ( A π * ) , surf ( A π * ) , and τ ( A π * ) for general π . The results established in Section 3, however, make it possible to derive simple expressions for both quantities.

We start with a simple illustrative example and then prove the general result (in a different manner).

Example 4.2

Consider N = 3 and the permutation π = ( 1 , 3 , 2 ) . Since, firstly, A π * C mcd , secondly, A π * is globally attractive, and since, thirdly, lim n τ ( V π n ( M ) ) = τ ( A π * ) implies lim n λ 2 ( Ω 2 V π n ( M ) ) = λ 2 ( Ω 2 A π * ) , it suffices to calculate λ 2 ( Ω 2 A π * ) , which can be done as follows: Obviously, we have (see Figure 2 for an illustration of steps 3–6 in the construction)

λ 2 ( Ω 2 V π ( M ) ) = 1 9 λ 2 ( Ω 2 V π 2 ( M ) ) = 1 9 + 3 1 9 2 = 1 9 1 + 1 3 λ 2 ( Ω 2 V π 3 ( M ) ) = 1 9 + 3 1 9 2 + 9 1 2 7 2 = 1 9 1 + 1 3 + 1 3 2 = λ 2 ( Ω 2 V π n ( M ) ) = 1 9 1 + 1 3 + + 1 3 n 1 ,

which yields

λ 2 ( Ω 2 A π * ) = 1 9 1 1 1 3 = 1 6 .

Having that, using equations (6), (10), and (14) shows

( A π * ) = 1 ( 2 2 ) 1 6 , τ ( A π * ) = 1 4 1 6 = 1 3 ,

as well as

surf ( A π * ) = 2 ( 2 2 1 3 ) 1 6 .

Figure 2 
               Supports of the copulas 
                     
                        
                        
                           
                              
                                 V
                              
                              
                                 π
                              
                              
                                 n
                              
                           
                           
                              (
                              
                                 M
                              
                              )
                           
                        
                        {V}_{\pi }^{n}\left(M)
                     
                   (black line segments) and the corresponding sets 
                     
                        
                        
                           
                              
                                 Ω
                              
                              
                                 0
                              
                              
                                 
                                    
                                       V
                                    
                                    
                                       π
                                    
                                    
                                       n
                                    
                                 
                                 
                                    (
                                    
                                       M
                                    
                                    )
                                 
                              
                           
                        
                        {\Omega }_{0}^{{V}_{\pi }^{n}\left(M)}
                     
                   (magenta squares), and 
                     
                        
                        
                           
                              
                                 Ω
                              
                              
                                 
                                    
                                       2
                                    
                                 
                              
                              
                                 
                                    
                                       V
                                    
                                    
                                       π
                                    
                                    
                                       n
                                    
                                 
                                 
                                    (
                                    
                                       M
                                    
                                    )
                                 
                              
                           
                        
                        {\Omega }_{\sqrt{2}}^{{V}_{\pi }^{n}\left(M)}
                     
                   (green squares) for 
                     
                        
                        
                           n
                           ∈
                           
                              {
                              
                                 3
                                 ,
                                 4
                                 ,
                                 5
                                 ,
                                 6
                              
                              }
                           
                        
                        n\in \left\{3,4,5,6\right\}
                     
                   and 
                     
                        
                        
                           π
                           =
                           
                              (
                              
                                 1
                                 ,
                                 3
                                 ,
                                 2
                              
                              )
                           
                        
                        \pi =\left(1,3,2)
                     
                   as considered in Example 4.2.
Figure 2

Supports of the copulas V π n ( M ) (black line segments) and the corresponding sets Ω 0 V π n ( M ) (magenta squares), and Ω 2 V π n ( M ) (green squares) for n { 3 , 4 , 5 , 6 } and π = ( 1 , 3 , 2 ) as considered in Example 4.2.

Theorem 4.3

Suppose that N 2 and that π is a permutation of { 1 , , N } . Then, the following identities hold for the copula A π * with self-similar support:

(21) τ ( A π * ) = 1 4 N N 1 λ 2 ( Ω 2 V π ( M ) ) = 1 4 1 N ( N 1 ) # { ( i , j ) { 1 , , N } 2 : π ( i ) < j and π 1 ( j ) < i }

(22) ( A π * ) = 1 ( 2 2 ) N N 1 λ 2 ( Ω 2 V π ( M ) ) = 1 ( 2 2 ) 1 N ( N 1 ) # { ( i , j ) { 1 , , N } 2 : π ( i ) < j and π 1 ( j ) < i }

(23) surf ( A π * ) = 2 ( 2 2 1 3 ) N N 1 λ 2 ( Ω 2 V π ( M ) )

Proof

First, note that for every A h C mcd , we have

(24) λ 2 ( Ω 2 V π ( A h ) ) = λ 2 ( Ω 2 V π ( M ) ) + 1 N λ 2 ( Ω 2 A h ) .

Since A π * = A h * for some λ -preserving bijection h * and since V π ( A π * ) = A π * holds, equation (24) implies

λ 2 ( Ω 2 A π * ) = λ 2 ( Ω 2 V π ( M ) ) + 1 N λ 2 ( Ω 2 A π * ) ,

from which we conclude

λ 2 ( Ω 2 A π * ) = N N 1 λ 2 ( Ω 2 V π ( M ) ) .

Having this, the desired identities follow by applying equations (6), (10), and (14).□

We conclude the article with the following example.

Example 4.4

Consider N = 4 and the permutation π = ( 3 , 1 , 4 , 2 ) . Figure 3 depicts the first four steps in the construction process of the corresponding copula A π * . In this case, we have

λ 2 ( Ω 2 V π ( M ) ) = 3 16 ,

and applying Theorem 4.3 directly yields

τ ( A π * ) = 0 , ( A π * ) = 1 2 + 2 4

and

surf ( A π * ) = 2 2 + 1 4 + 3 4 .

Figure 3 
               Supports of the copulas 
                     
                        
                        
                           
                              
                                 V
                              
                              
                                 π
                              
                              
                                 n
                              
                           
                           
                              (
                              
                                 M
                              
                              )
                           
                        
                        {V}_{\pi }^{n}\left(M)
                     
                   (black line segments) and the corresponding sets 
                     
                        
                        
                           
                              
                                 Ω
                              
                              
                                 0
                              
                              
                                 
                                    
                                       V
                                    
                                    
                                       π
                                    
                                    
                                       n
                                    
                                 
                                 
                                    (
                                    
                                       M
                                    
                                    )
                                 
                              
                           
                        
                        {\Omega }_{0}^{{V}_{\pi }^{n}\left(M)}
                     
                   (magenta) and 
                     
                        
                        
                           
                              
                                 Ω
                              
                              
                                 
                                    
                                       2
                                    
                                 
                              
                              
                                 
                                    
                                       V
                                    
                                    
                                       π
                                    
                                    
                                       n
                                    
                                 
                                 
                                    (
                                    
                                       M
                                    
                                    )
                                 
                              
                           
                        
                        {\Omega }_{\sqrt{2}}^{{V}_{\pi }^{n}\left(M)}
                     
                   (green) for 
                     
                        
                        
                           n
                           ∈
                           
                              {
                              
                                 1
                                 ,
                                 2
                                 ,
                                 3
                                 ,
                                 4
                              
                              }
                           
                        
                        n\in \left\{1,2,3,4\right\}
                     
                   and 
                     
                        
                        
                           π
                           =
                           
                              (
                              
                                 3
                                 ,
                                 1
                                 ,
                                 4
                                 ,
                                 2
                              
                              )
                           
                        
                        \pi =\left(3,1,4,2)
                     
                   as considered in Example 4.4.
Figure 3

Supports of the copulas V π n ( M ) (black line segments) and the corresponding sets Ω 0 V π n ( M ) (magenta) and Ω 2 V π n ( M ) (green) for n { 1 , 2 , 3 , 4 } and π = ( 3 , 1 , 4 , 2 ) as considered in Example 4.4.

5 Conclusion and outlook

As main results of the article, we have established affine interrelations between Kendall’ τ , the length profile (as introduced in [1]), and the surface area of mutually completely dependent bivariate copulas. In addition, we have shown that these interrelations do not carry over to the full class C of all bivariate copulas, in fact they do not even hold for the product copula Π . A small application of the derived identities to the calculation of Kendall’s τ , the length of measure, and the surface area of mutually completely dependent copulas with self-similar support rounds off the article.

Related natural open questions, which we plan to tackle in the future, are, in particular, the following two:

  1. Is it possible to extend equation (14) linking Kendall’s τ and surf to the multivariate setting? Note that for d -dimensional completely dependent copulas with d 3 (see, e.g., [7] for a definition and some properties), the integrand of the surface integral is a step function again, which will, however, attain up to d + 1 distinct values, so expressing the surface area solely in terms of the Lebesgue measure of one single set will most likely not be possible.

  2. According to Remark 3.9, the two most well known concordance measures can (at least on a dense subclass) be interpreted geometrically in terms of surface and volume, respectively. It remains to be analyzed if other concordance measures allow for nice geometric interpretations as well.

Acknowledgments

The second author gratefully acknowledge the support of the WISS 2025 project “IDA-lab Salzburg” (20204-WISS/225/197-2019 and 20102-F1901166-KZP).

  1. Conflict of interest: The authors state that there is no conflict of interest.

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Received: 2023-03-27
Revised: 2023-08-29
Accepted: 2023-09-22
Published Online: 2023-11-10

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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