Rank dependence¶
Throughout this chapter, \(C\) is a bivariate copula, \((U,V)\sim C\), and \(K_C(u,t)\) is a version of \(\mathbb P(V\leq t\mid U=u)\). These are population coefficients. No sample-ranking procedure or numerical estimator is implicit in the definitions.
Six coefficients¶
| Coefficient | Definition | Range |
|---|---|---|
| Spearman's rho | \(\rho(C)=12\int uv\,dC(u,v)-3\) | \([-1,1]\) |
| Kendall's tau | \(\tau(C)=4\int C(u,v)\,dC(u,v)-1\) | \([-1,1]\) |
| Spearman's footrule | \(\phi(C)=6\int_0^1 C(t,t)\,dt-2\) | \([-\tfrac12,1]\) |
| Gini's gamma | \(\gamma(C)=4\int_0^1[C(t,t)+C(t,1-t)]\,dt-2\) | \([-1,1]\) |
| Blomqvist's beta | \(\beta(C)=4C(\tfrac12,\tfrac12)-1\) | \([-1,1]\) |
| Chatterjee's xi | \(\xi(C)=6\int_0^1\int_0^1 K_C(u,t)^2\,du\,dt-2\) | \([0,1]\) |
The footrule convention is the normalized population copula functional. Chatterjee's xi measures dependence of the second coordinate on the first; transposing the copula reverses that direction.
Formal definitions of rho, tau, footrule, gamma, and beta
For the full range proofs and conditional-kernel definition of xi, see the rank API guide.
Benchmarks¶
| Copula | \(\rho\) | \(\tau\) | \(\phi\) | \(\gamma\) | \(\beta\) | \(\xi\) |
|---|---|---|---|---|---|---|
| Independence \(\Pi\) | 0 | 0 | 0 | 0 | 0 | 0 |
| Comonotonicity \(M\) | 1 | 1 | 1 | 1 | 1 | 1 |
| Countermonotonicity \(W\) | −1 | −1 | −1/2 | −1 | −1 | 1 |
Both \(V=U\) and \(V=1-U\) determine the second coordinate completely. That is why xi equals one in both cases, although their concordance signs differ.
What do extreme values identify?¶
Extremes of Spearman's rho
Kendall's tau has analogous equality cases; for example:
Maximal beta does not characterize \(M\). It characterizes equal-split ordinal sums, whose two components can vary. The ordinal-sum chapter explains this distinction.
Detecting independence¶
Xi vanishes exactly at independence
For every bivariate copula, including singular ones,
Zero rho or zero tau alone does not characterize independence. Under positive quadrant dependence, however, zero rho does:
Zero rho in the PQD class
If \(C(u,v)\geq uv\) for every \((u,v)\in I^2\), then
The rank guide also describes the NQD analogues, Kendall's concordance-probability interpretation, reflection identities, mixture formulas, and the remaining equality cases.
Library revision: fe53ea2f · Lean 4.34.0