On the exact region determined by Kendall's τ and Spearman's ρ
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.
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