# On the exact region determined by Kendall’s τ and Spearman’s ρ [by Wolfgang Trutschnig]

Suppose that ${X,Y}$ are random variables with continuous distribution functions ${F}$ and ${G}$ respectively. Then Spearman’s ${\rho}$ is defined as the Pearson correlation coefficient of the ${\mathcal{U}(0,1)}$-distributed random variables ${U:=F \circ X}$ and ${V:=G \circ Y}$, and Kendall’s ${\tau}$ is given by the probability of concordance minus the probability of discordance, i.e.

$\displaystyle \rho(X,Y)=12\big(\mathbb{E}(UV)-\tfrac{1}{4}\big)$

$\displaystyle \tau(X,Y)= \mathbb{P}\big((X_1-X_2)(Y_1 -Y_2)>0) - \mathbb{P}\big((X_1-X_2)(Y_1 -Y_2)<0\big)$

where ${(X_1,Y_1)}$ and ${(X_2,Y_2)}$ are independent and have the same distribution as ${(X,Y)}$. Clearly, ${\tau}$ and ${\rho}$ are the two most famous nonparametric measures of concordance. Despite the fact that ${\tau}$ and ${\rho}$ are both concordance measures, they quantify different aspects of the underlying dependence structure and may vary significantly. Since the early 1950s two universal inequalities between ${\tau}$ and ${\rho}$ are known – the first one goes back to Daniels (1950), the second one to Durbin and Stuart (1951):

$\displaystyle \vert 3 \tau-2 \rho \vert \leq 1 \ \ \ \ \ (1)$

$\displaystyle \frac{(1+\tau)^2}{2} -1 \leq \rho \leq 1- \frac{(1-\tau)^2}{2} \ \ \ \ \ (2)$

Although both inequalities have been known for decades now, a full characterization of the exact ${\tau}$${\rho}$-region ${\Omega}$, defined by

$\Omega=\big\{(\tau(X,Y),\rho(X,Y)):\,\, X,Y \text{ continuous random variables}\big\},$

was first established in our article. Working with copulas and so-called shuffles it was possible to describe all random variables ${(X,Y)}$ with ${(\tau(X,Y),\rho(X,Y)) \in \partial \Omega}$, where ${\partial \Omega}$ denotes the topological boundary of ${\Omega}$. Figure 0 depicts some distributions of uniformly distributed random variables ${X,Y}$ for which we have ${(\tau(X,Y),\rho(X,Y)) \in \partial \Omega}$ – for a small animation we refer to animation ${\partial \Omega}$. Although our main objective was a full characterization of ${\Omega}$ the proofs produced an equally interesting by-product: For every point ${(x,y) \in \Omega}$ there exist random variables ${X,Y}$ and a measurable bijection ${f: \mathbb{R} \rightarrow \mathbb{R}}$ with ${Y=f(X)}$ such that ${(\tau(X,Y),\rho(X,Y))=(x,y)}$ holds. In other words: (Mutually) completely dependent random variables cover all possible ${\tau}$${\rho}$-constellations.

As pointed out in Section 6 in our paper, characterizing the exact ${\tau}$${\rho}$-region for standard families of distributions (or, equivalently, copulas) may in some cases be even more difficult than determining ${\Omega}$ was. The main reason for this fact is that not in all families we may find dense subsets consisting of elements for which ${\tau}$ and ${\rho}$ allow for handy formulas (as it is the case for shuffles). The author conjectures, however, that the classical Hutchinson-Lai inequalities are not sharp for the class of all extreme-value copulas and that it might be possible to derive sharper inequalities in the near future.