On the exact region determined by Kendall's tau and Spearman's rho

TL;DR

This paper precisely characterizes the region of all possible pairs of Kendall's tau and Spearman's rho, revealing its geometric properties and the nature of dependence structures that realize these values.

Contribution

It provides an exact description of the region determined by Kendall's tau and Spearman's rho, including its shape, boundary, and the dependence structures that attain these values.

Findings

The inequality by Durbin and Stuart is sharp only on a countable set.

The region is compact, simply connected, but not convex.

Every point in the region corresponds to mutually completely dependent variables.

Abstract

Using properties of shuffles of copulas and tools from combinatorics we solve the open question about the exact region $\Omega$ determined by all possible values of Kendall's $\tau$ and Spearman's $\rho$. In particular, we prove that the well-known inequality established by Durbin and Stuart in 1951 is only sharp on a countable set with sole accumulation point $(-1,-1)$, give a simple analytic characterization of $\Omega$ in terms of a continuous, strictly increasing piecewise concave function, and show that $\Omega$ is compact and simply connected but not convex. The results also show that for each $(x,y)\in \Omega$ there are mutually completely dependent random variables whose $\tau$ and $\rho$ values coincide with $x$ and $y$ respectively.