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Kendall's tau of an Archimedean copula via its generator

Nelsen, An Introduction to Copulas, second edition, Corollary 5.1.4: for an Archimedean copula with generator φ, τ_C = 1 + 4 ∫₀¹ φ(t) / φ'(t) dt.

We prove this for generators whose inverse generator ψ has a continuous derivative where it is positive (BivariateGenerator.IsC1; strict and non-strict generators alike), in the form τ_C = 1 + 4 ∫₀¹ φ(t) ψ'(φ(t)) dt (BivariateGenerator.IsC1.kendallTau_eq) and in Nelsen's form with the derivative of the generator (BivariateGenerator.IsC1.kendallTau_eq_deriv).

As in Nelsen's proof, τ_C = 4 E[C(U, V)] − 1 with E[C(U, V)] = ∫₀¹ (1 − K_C(t)) dt (layer-cake formula), and the Kendall distribution function K_C(t) = t − φ(t) / φ'(t) of Copula.Archimedean.KendallDistribution (Nelsen, Theorem 4.3.4).

For the outer power φ^δ, δ ≥ 1 (Nelsen, Section 4.5, the family C_{φ^δ}), the integrand scales by 1/δ, so τ_{φ^δ} = 1 + (τ_φ − 1) / δ (BivariateGenerator.IsC1.kendallTau_outerPower).

theorem ProbabilityTheory.Copula.BivariateGenerator.IsC1.kendallTau_eq {g : BivariateGenerator} {ψ' : ℝ → ℝ} (h : g.IsC1 ψ') : g.copula.kendallTau = 1 + 4 * ∫ (t : ℝ) in 0..1, g.invFunReal t * ψ' (g.invFunReal t)

Nelsen, Corollary 5.1.4: τ_C = 1 + 4 ∫₀¹ φ(t) ψ'(φ(t)) dt for a C¹ generator, where φ(t) ψ'(φ(t)) = φ(t) / φ'(t).

theorem ProbabilityTheory.Copula.BivariateGenerator.IsC1.kendallTau_eq_deriv {g : BivariateGenerator} {ψ' : ℝ → ℝ} (h : g.IsC1 ψ') : g.copula.kendallTau = 1 + 4 * ∫ (t : ℝ) in 0..1, g.invFunReal t / deriv g.invFunReal t

Nelsen, Corollary 5.1.4 in its original form τ_C = 1 + 4 ∫₀¹ φ(t) / φ'(t) dt, with φ' the derivative of the generator.

theorem ProbabilityTheory.Copula.BivariateGenerator.IsC1.outerPower {g : BivariateGenerator} {ψ' : ℝ → ℝ} (h : g.IsC1 ψ') (δ : ℝ) (hδ : 1 ≤ δ) : (g.outerPower δ hδ).IsC1 fun (s : ℝ) => ψ' (s ^ δ⁻¹) * (δ⁻¹ * s ^ (δ⁻¹ - 1))

The outer power φ^δ of a C¹ generator is C¹, with (ψ(s^{1/δ}))' = ψ'(s^{1/δ}) · s^{1/δ - 1} / δ.

theorem ProbabilityTheory.Copula.BivariateGenerator.outerPower_integrand (g : BivariateGenerator) (ψ' : ℝ → ℝ) (δ : ℝ) (hδ : 1 ≤ δ) {t : ℝ} (ht : t ∈ Set.Ioc 0 1) : (g.outerPower δ hδ).invFunReal t * (ψ' ((g.outerPower δ hδ).invFunReal t ^ δ⁻¹) * (δ⁻¹ * (g.outerPower δ hδ).invFunReal t ^ (δ⁻¹ - 1))) = δ⁻¹ * (g.invFunReal t * ψ' (g.invFunReal t))

The Kendall integrand φ ψ'(φ) of the outer power φ^δ is 1/δ times that of φ.

theorem ProbabilityTheory.Copula.BivariateGenerator.IsC1.kendallTau_outerPower {g : BivariateGenerator} {ψ' : ℝ → ℝ} (h : g.IsC1 ψ') (δ : ℝ) (hδ : 1 ≤ δ) : (g.outerPower δ hδ).copula.kendallTau = 1 + (g.copula.kendallTau - 1) / δ

Kendall's tau of the outer power φ^δ: τ_{φ^δ} = 1 + (τ_φ − 1) / δ.