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)

{The classical τ-ρ-region determined by the inequalities (1) and (2) and some copulas (distributing mass uniformly on the blue segments) for which the inequality by Durbin and Stuart is sharp. The red line depicts the true boundary of Ω.

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.

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