On the exact region determined by Kendall'sτand Spearman'sρ
Manuela Schreyer, Roland Paulin, Wolfgang Trutschnig
Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2017 · Volume 79 · Issue 2
Abstract
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'sτand Spearman'sρ. 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 variablesXandYwhoseτ- andρ-values coincide withxandyrespectively.
Keywords
Concordance, Copula, Kendallτ, Shuffle, Spearmanρ
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