On the Exact Regions Determined by Kendall’s Tau and Other Concordance Measures

We determine the upper and lower bounds for possible values of Kendall’s tau of a bivariate copula given that the value of its Spearman’s footrule or Gini’s gamma is known, and show that these bounds are always attained.


Introduction
One of the most important notions in statistics is the notion of dependence of random variables. When we measure the dependence, we often try to describe it with a single real number. The most commonly used measure is Pearson's correlation coefficient, which measures linear dependence. For the random pair (X, Y ) Pearson's correlation coefficient depends not only on the degree of association between X and Y but also on the marginal distributions of the pair (X, Y ). If we want to measure only the degree of association, we need measures that do not depend on the marginals of the random vector, but only on the copula connecting its components. This is often done with the help of a concordance measure, or its generalization, weak concordance measure.
Intuitively, two continuous random variables X and Y are in concordance when large values of X occur simultaneously with large values of Y . More precisely, two realizations (x 1 , y 1 ) and (x 2 , y 2 ) of the random vector (X, Y ) are concordant when (x 2 − x 1 )(y 2 − y 1 ) > 0 and they are discordant when (x 2 − x 1 )(y 2 − y 1 ) < 0. We can measure the concordance of a pair of random variables (X, Y ) in various ways, see [26]. The most commonly used concordance measures are Spearman's rho, Kendall's tau, Gini's gamma and for any copula C, and in Sect. 4, we show that the exact region between Kendall's tau and Gini's gamma is determined by the inequalities In both cases the bounds are attained. In Sect. 5, we give the similarity measure between Kendall's tau and other (weak) concordance measures.

Preliminaries
Let I = [0, 1] be the unit interval and R = [u 1 , u 2 ] × [v 1 , v 2 ] a rectangle contained in I 2 with u 1 u 2 and v 1 v 2 . Given a real function C : . A (bivariate) copula is a function C : I 2 → I with the following properties: (i) C(0, v) = C(u, 0) = 0 for all u, v ∈ I (C is grounded), (ii) C(u, 1) = u and C (1, v) = v for all u, v ∈ I (C has uniform marginals), and (iii) V C (R) 0 for every rectangle R ⊆ I 2 (C is 2−increasing). Any copula C induces a probability measure μ C on the Borel σ-algebra in I 2 . This measure is uniquely determined by the property μ C (R) = V C (R) for all rectangles R ⊆ I 2 . Furthermore, measure μ C is bistochastic in the sense that μ C (B × I) = μ C (I × B) = λ(B) for any Borel set B ⊆ I, where λ denotes the Lebesgue measure on I. The set of all bivariate copulas will be denoted by C. It is well known that this set is compact in the sup norm. For any copula C its diagonal will be denoted by δ C , i.e., δ C (u) = C(u, u) for all u ∈ I.
Given two copulas C and D, we denote This is the so-called pointwise order of copulas. For any copula C, we have W C M , where W (u, v) = max{0, u + v − 1} and M (u, v) = min{u, v} are the lower and upper Fréchet-Hoeffding bounds for the set of all copulas. Furthermore, any copula C induces reflected copulas C σ1 and C σ2 Let h : I → I be a measure preserving bijection, where I is equipped with the Lebesgue measure λ. Then, the function defined by is a copula whose mass in concentrated on the graph of h, i.e., . . u n−1 u n = 1, shortly written as (n − 1)-tuple of splitting points J = (u 1 , u 2 , . . . , u n−1 ), a permutation π ∈ S n , written as n-tuple of images π = (π(1), π(2), . . . , π(n)), and a mapping ω : {1, 2, . . . , n} → {−1, 1}, written as n-tuple of images ω = (ω(1), ω(2), . . . , ω(n)). The mass of C is concentrated on the diagonals of the squares Hence, C is defined by the measure preserving bijection h C : I → I given by Furthermore, C is the copula of uniformly distributed random variables U and V on I with the property P (V = h C (U )) = 1. For more details see [23,Sect. 3.2.3].
In [26], Scarsini introduced formal axioms for concordance measures. These are mappings that assign to each copula a real number in [−1, 1] and are meant to measure the degree of concordance/discordance between the components of random vectors. Recall that two observations (x, y) and (x , y ) For the formal definition of concordance measures and further details, we refer the reader to [6,23]. Here, we give the properties of concordance measures, which we will need in the sequel: if κ is a concordance measure, then κ(M ) = 1, κ(C σ1 ) = κ(C σ2 ) = −κ(C) for any copula C ∈ C, κ is continuous with respect to the pointwise convergence, and κ is monotone increasing with respect to the pointwise order.
Many of the most important bivariate concordance measures, including Kendall's tau and Gini's gamma, can be expressed with the so-called concordance function Q, introduced by Kruskal [19]. If (X 1 , Y 1 ) and (X 2 , Y 2 ) are pairs of continuous random variables, then the concordance function of random vectors (X 1 , Y 1 ) and (X 2 , Y 2 ) depends only on the corresponding copulas C 1 and C 2 and is given by (see [23,Theorem 5 It turns out that the concordance function is symmetric in its arguments, i.e., Q(C 1 , C 2 ) = Q(C 2 , C 1 ), and has several other useful properties, see [ and Gini's gamma by In [20], Liebscher considered weak concordance measures, which are slightly more general mappings than concordance measures (the formal definition can be found in Liebscher's paper). The most important example of a weak concordance measure is Spearman's footrule defined by Spearman's footrule is not a true concordance measure since its range is There is an abundance of information in the literature on all three (weak) concordance measures defined above, including discussions on their statistical meaning. Kendall's tau was investigated in [11,13,14,31], Gini's gamma in [2,12,22], and Spearman's footrule in [4,12,28,30].
Connections between different (weak) concordance measures were investigated in [9,[16][17][18]27]. Here we only give the results which include Kendall's tau for the sake of completeness. For any copula C ∈ C we have and the bounds are attained (see [23]). The bounds for Kendall's tau with respect to Spearman's rho are given by for some n ≥ 2, and for every n ∈ N, n 2, Φ n : The bounds are attained, see [27]. Figure 1 depicts the exact regions determined by Spearman's rho and Kendall's tau and by Blomqvist's beta and Kendall's tau.

The Exact Region Determined by τ and φ
In this section we will describe the exact region determined by Kendall's tau and Spearman's footrule.

Proposition 1.
Let h : I → I be a bijective measure preserving function. Let C ∈ C be a copula defined by (1) with the mass concentrated on the graph of h, i.e. μ C ({(u, h(u)), u ∈ I}) = 1. Then We introduce a new variable t = h(u) into the second to last integral in (6) to get since h is bijective and measure preserving. By looking at the mass of the copula C inside the rectangle from the origin to the point (max{u, h(u)}, max{u, h(u)}), we obtain Notice that f (t, u) + f (u, t) = 1 almost everywhere on I 2 , since where λ 2 is the Lebesgue measure on I 2 . It follows that the last integral in (6) equals We finally obtain that The following example gives copulas for which the bound of Proposition 1 is attained. The scatterplot of copula A a is shown in Fig. 2 (left). Notice that A 0 = W and A 1 = M . We have In next example, we give copulas that will correspond to the points on the lower bound of the region determined by Kendall's tau and Spearman's footrule.
The scatterplot of copula B a is shown in Fig. 2 (right). Notice that B 1 = M . We have  ; C ∈ C} is a triangular region given by Proof. We will first prove that the lower bound from Proposition 1 holds for any copula C.
Let ε > 0. Since τ is a concordance measure, there exists δ > 0 such that for every copula C with sup (u,v) by Proposition 1. By sending ε to 0, we obtain the desired lower bound. Next we prove the upper bound. For any copula C we can estimate τ (C) = 4 Since the concordance function is symmetric we obtain τ (C) 4 So, any copula C satisfies the given inequalities and all the points on the upper and lower boundary of the region are attained by copulas from Examples 2 and 3. It remains to be shown that all the points in between are attained as well.

The Exact Region Determined by τ and γ
In this section, we will describe the exact region determined by Kendall's tau and Gini's gamma. We first provide examples of copulas that will correspond to the points on the boundary of the region. The scatterplot of copulas C a and D a are shown in Fig. 4. We have