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

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 .


Introduction
Spearman's ρ of a bivariate copula A is a rescaled version of the volume below the graph of A (see [3,11]) in the sense that A dλ 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 which lead the authors of [1] to conjecturing 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], 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 afore-mentioned facts, the objective of this note is two-fold: 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., that 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 a simple geometric identity linking the length measure and the surface area with the area of the set Ω √ 2 , given by where h denotes the transformation corresponding to the completely dependent copula A h , will be key.An application to calculating Kendall's τ , the length measure and the surface area of completely dependent copulas with self-similar support concludes the paper.

Notation 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 ) is a compact metric space and that in C d pointwise and uniform convergence are equivalent.For more background on copulas and doubly stochastic measures we refer to [3,11].
In what follows, Markov kernels will be a handy tool.A mapping ) is measurable for every fixed B ∈ B(R) and the mapping B → K(x, B) is a probability measure for every fixed x ∈ R. A Markov kernel K : R × B(R) → [0, 1] is called regular conditional distribution of a (real-valued) random variable Y given (another random variable) X if for every B ∈ B(R) holds P-a.s.It is well known that a regular conditional distribution of Y given X exists and is unique P X -almost surely.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 . Note that for every A ∈ C and Borel sets E, F ∈ B([0, 1]) we have the following disintegration formulas: For more details and properties of conditional expectations and regular conditional distributions we refer to [8,10].
A copula A ∈ C will be called completely 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 , the class of all completely dependent copulas by C cd .A completely dependent copula A h is called mutually completely dependent, if the transformation h is bijective.Notice that in this case the transpose A t h of A h , defined by A t h (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 .It is well known (see [3,11]) that C mcd is dense in (C, d ∞ ), in fact even the family of all equidistant even shuffles (again see [3,11]) is dense.For further properties of completely dependent copulas we refer to [13] and the references therein.
Turning towards the length profile introduced and studied in [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's arc-length.Then the length profile of A is defined as the function (3) holds for every t ∈ (0, 1).Building upon L A the so-called length measure (A) of A is defined as and describes the average arc-length of upper t-cuts of A.
as well as (A) ∈ [ 1 √ 2 , 1] holds (ineq.( 5) was also one of the reasons for falsely conjecturing that the length measure might be transformable into a concordance measure).
In [1] it was shown 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 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 with Throughout the rest of this note we will only write Ω √ 2 instead of Ω A h √ 2 whenever no confusion will arise.Notice that for classical equidistant straight shuffles eq. ( 6) implies that (A h ) can be calculated by simply counting squares as Figure 2 illustrates in terms of two simple examples -one shuffle with three, and a second one with nice equidistant stripes.
for the first shuffle and for the second one.

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 [11]) and finally applying an approximation result like [9, Theorem 3.2] yields that for arbitrary bivariate copulas A, B ∈ C the following identity holds: For A h ∈ C mcd eq. ( 8) 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 Using disintegration and change of coordinates directly yields and hence proves eq. ( 8).The latter identity, however, boils down to an affine transformation of λ 2 (Ω √ 2 ) by considering Having this, the identity follows immediately.Notice that eq. ( 9) implies that the area of Ω √ 2 coincides with the quantity inv(h) as studied in [12,Lemma 3.1].Comparing eq. ( 6) and eq.( 9) shows the existence of an affine transformation a : 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 measure and Kendall's τ holds: Theorem 3.1 provides an 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 [1] it was further shown that the length measure interpreted as function : 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: . .are mutually completely dependent copulas and that the sequence (A hn ) 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 hn ) = τ (A h ) and eq.( 9) directly yields lim n→∞ (A hn ) = (A h ).Proof.According to [12] 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 eq.( 9).
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 by surf 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 by Proof.For the case of a completely dependent copula A h eq. ( 11) obviously simplifies to Considering that the latter integrand is a step function only attaining the values 1, √ 2 and √ 3, defining as well as (Ω √ 2 as before) The latter identity can be further simplified: The measurable bijection Therefore using the fact that 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: As in the case of the length measure we have the following two immediate corollaries: Corollary 3.6.The mapping surf : ] there exists some mutually completely copula A h with surf(A h ) = z.In other words, all values in ] are attained by surf.
Remark 3.8.The afore-mentioned 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 [3,4,12] (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 [12] considering all shuffles is sufficient) is therefore reminiscent of the famous isoperimetric inequality bounding the surface area of a set by a function of the volume (see [6]).
4 Calculating τ, and surf for mutually completely dependent copulas with self-similar support 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 measure of copulas of this type.
According to [13] for every transformation matrix T there exists a unique copula 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 ij ) i=1...N, j=1...N , defined by is obviously a transformation matrix.To simplify notation we will simply write V π := V Tπ as well as A * Tπ = 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 [13]) C cd is closed in (C, D 1 ) using eq.( 16) 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 the previous section, 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).Having that, using eqs.( 6), ( 9) and ( 13