The monotone part of Chatterjee's rank correlation

Source abstract

Chatterjee's rank correlation $ξ$ measures how strongly a random variable $Y$ depends functionally on a random variable $X$, but it does not distinguish increasing, decreasing and non-monotone relationships. We ask how much of the dependence measured by $ξ$ is explained by an increasing relationship. For continuous marginals, $ξ$ is, up to a factor, a squared distance from independence in a Hilbert space geometry derived from the Sobolev scalar product for copulas. We answer it by the metric projection onto the closed convex set of stochastically increasing copulas in this geometry. The projection is explicit: it replaces every vertical section by its least concave majorant, and it coincides with the smallest stochastically increasing copula dominating the given one. It splits $ξ$ orthogonally into a monotone part $ξ^{\uparrow}$ and the squared distance to the class of stochastically increasing copulas; dually, $ξ^{\uparrow}$ is, up to a factor, the squared distance to the polar cone, which corresponds to negative quadrant dependence. The monotone part vanishes exactly for negatively quadrant dependent copulas, equals $ξ$ exactly for stochastically increasing copulas, and equals one only for the comonotonicity copula. In contrast to $ξ$, it is increasing in the concordance order and continuous with respect to weak convergence. Symmetrisation decomposes the measure of mutual complete dependence in the same way. Finally, the construction extends to arbitrary distributions via isotonic regression of conditional distribution functions. In particular, the plug-in estimator based on the empirical distribution is strongly consistent for every copula, without smoothing or tuning parameters.