← Copula mathematical handbook

Affine dependence coefficients under convex mixtures

Rho, footrule, gamma and beta are affine in the copula. This statement does not extend to Kendall's tau or Chatterjee's xi.

theorem ProbabilityTheory.Copula.spearmanRho_mix (C D : Copula 2) (a : ↑unitInterval) : (C.mix D a).spearmanRho = ↑a * C.spearmanRho + (1 - ↑a) * D.spearmanRho

theorem ProbabilityTheory.Copula.spearmanFootrule_mix (C D : Copula 2) (a : ↑unitInterval) : (C.mix D a).spearmanFootrule = ↑a * C.spearmanFootrule + (1 - ↑a) * D.spearmanFootrule

theorem ProbabilityTheory.Copula.giniGamma_mix (C D : Copula 2) (a : ↑unitInterval) : (C.mix D a).giniGamma = ↑a * C.giniGamma + (1 - ↑a) * D.giniGamma

theorem ProbabilityTheory.Copula.blomqvistBeta_mix (C D : Copula 2) (a : ↑unitInterval) : (C.mix D a).blomqvistBeta = ↑a * C.blomqvistBeta + (1 - ↑a) * D.blomqvistBeta