On the Exact Region Determined by Kendall's τ and Spearman's ρ

Summary Using properties of shuffles of copulas and tools from combinatorics we solve the open question about the exact region Ω determined by all possible values of Kendall's τ and Spearman's ρ. In particular, we prove that the well-known inequality established by Durbin and Stuart in 1951 is not sharp outside a countable set, give a simple analytic characterization of Ω in terms of a continuous, strictly increasing piecewise concave function and show that Ω is compact and simply connected, but not convex. The results also show that for each (x, y) ∈ Ω there are mutually completely dependent random variables X and Y whose τ- and ρ-values coincide with x and y respectively.