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Table 6: signed Clayton Kendall tau and positive Clayton Chatterjee xi

theorem Papers.AnsariRockel2024.clayton_positive_conditionalCDF {θ : ℝ} (hθ : 0 < θ) (v : ↑unitInterval) (hv : 0 < ↑v) : (fun (u : ↑unitInterval) => (ProbabilityTheory.Copula.clayton 2 θ hθ).conditionalCDF u v) =ᵐ[MeasureTheory.volume] fun (u : ↑unitInterval) => Verification.claytonPartial θ ↑u ↑v

theorem Papers.AnsariRockel2024.clayton_positive_kendallTau {θ : ℝ} (hθ : 0 < θ) : (ProbabilityTheory.Copula.clayton 2 θ hθ).kendallTau = θ / (θ + 2)

theorem Papers.AnsariRockel2024.clayton_zero_kendallTau : (ProbabilityTheory.Copula.independence 2).kendallTau = 0

theorem Papers.AnsariRockel2024.clayton_negative_one_kendallTau : (ProbabilityTheory.Copula.claytonNegative (-1) ⋯ ⋯).kendallTau = -1

theorem Papers.AnsariRockel2024.clayton_negative_conditionalCDF {θ : ℝ} (hmin : -1 < θ) (hmax : θ < 0) (v : ↑unitInterval) (hv : 0 < ↑v) : (fun (u : ↑unitInterval) => (ProbabilityTheory.Copula.claytonNegative θ ⋯ hmax).conditionalCDF u v) =ᵐ[MeasureTheory.volume] fun (u : ↑unitInterval) => Verification.claytonNegativePartial θ ↑u ↑v

theorem Papers.AnsariRockel2024.clayton_negative_kendallTau {θ : ℝ} (hmin : -1 ≤ θ) (hmax : θ < 0) : (ProbabilityTheory.Copula.claytonNegative θ hmin hmax).kendallTau = θ / (θ + 2)

theorem Papers.AnsariRockel2024.clayton_positive_chatterjeeXi {θ : ℝ} (hθ : 0 < θ) : (ProbabilityTheory.Copula.clayton 2 θ hθ).chatterjeeXi = (6 * ∫ (v : ↑unitInterval), Verification.eulerHypergeometric (1 / θ) (2 + 2 / θ) (1 / θ + 1) (1 - ↑v ^ (-θ))) - 2