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Kendall's tau of mixtures

Tau is a quadratic functional, with cross terms given by the concordance function. In particular, dilution by independence involves both tau and rho.

theorem ProbabilityTheory.Copula.kendallTau_finiteMixture {n : ℕ} (C : Fin n → Copula 2) (w : Fin n → ℝ) (hw : ∀ (j : Fin n), 0 ≤ w j) (hsum : ∑ j : Fin n, w j = 1) : (finiteMixture C w hw hsum).kendallTau = ∑ i : Fin n, ∑ j : Fin n, w i * w j * (C i).concordanceQ (C j)

theorem ProbabilityTheory.Copula.kendallTau_mix (C D : Copula 2) (a : ↑unitInterval) : (C.mix D a).kendallTau = ↑a ^ 2 * C.kendallTau + (1 - ↑a) ^ 2 * D.kendallTau + 2 * ↑a * (1 - ↑a) * C.concordanceQ D

theorem ProbabilityTheory.Copula.kendallTau_mix_independence (C : Copula 2) (a : ↑unitInterval) : (C.mix (independence 2) a).kendallTau = ↑a ^ 2 * C.kendallTau + 2 / 3 * ↑a * (1 - ↑a) * C.spearmanRho

theorem ProbabilityTheory.Copula.kendallTau_mix_comonotonic (C : Copula 2) (a : ↑unitInterval) : (C.mix (comonotonic 2) a).kendallTau = ↑a ^ 2 * C.kendallTau + (1 - ↑a) ^ 2 + 2 / 3 * ↑a * (1 - ↑a) * (2 * C.spearmanFootrule + 1)

theorem ProbabilityTheory.Copula.kendallTau_mix_countermonotonic (C : Copula 2) (a : ↑unitInterval) : (C.mix countermonotonic a).kendallTau = ↑a ^ 2 * C.kendallTau - (1 - ↑a) ^ 2 + 2 * ↑a * (1 - ↑a) * (C.giniGamma - (2 * C.spearmanFootrule + 1) / 3)

theorem ProbabilityTheory.Copula.kendallTau_mix_comonotonic_countermonotonic (a : ↑unitInterval) : ((comonotonic 2).mix countermonotonic a).kendallTau = 2 * ↑a - 1

theorem ProbabilityTheory.Copula.kendallTau_not_affine : ¬∀ (C D : Copula 2) (a : ↑unitInterval), (C.mix D a).kendallTau = ↑a * C.kendallTau + (1 - ↑a) * D.kendallTau

Tau is not affine on the convex set of copulas.