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Kendall's tau of Archimedean families

Applications of Nelsen's Corollary 5.1.4 (BivariateGenerator.IsC1.kendallTau_eq), see Nelsen, An Introduction to Copulas, second edition, Section 5.1.1 and Table 4.1:

• Clayton (family 4.2.1), θ > 0: τ = θ / (θ + 2) (kendallTau_clayton);
• Gumbel–Hougaard (family 4.2.4), θ ≥ 1: τ = 1 − 1/θ (kendallTau_gumbel);
• BB1 (the outer power φ^δ of Clayton's generator, Nelsen Section 4.5): τ = 1 − 2 / (δ (θ + 2)) (kendallTau_bb1), via τ_{φ^δ} = 1 + (τ_φ − 1)/δ;
• Nelsen's families 4.2.12 and 4.2.14 (subfamilies of BB1): τ = 1 − 2/(3θ) (kendallTau_nelsen12) and τ = (2θ − 1)/(2θ + 1) (kendallTau_nelsen14);
• the non-strict family 4.2.2 (outer powers of the generator of W): τ = 1 − 2/θ (kendallTau_nelsen2).

Both generators are strict with continuously differentiable inverse generators (isC1_claytonGenerator, isC1_gumbelGenerator), and the integrands φ/φ' are elementary: −(t − t^{θ+1})/θ for Clayton and t log t / θ for Gumbel.

theorem ProbabilityTheory.Copula.integral_mul_log_unit : ∫ (x : ℝ) in 0..1, x * Real.log x = -1 / 4

∫₀¹ t log t dt = −1/4.

noncomputable def ProbabilityTheory.Copula.claytonGeneratorDeriv (θ s : ℝ) : ℝ

The derivative of Clayton's inverse generator (1 + s)^(-1/θ).

Equations

• ProbabilityTheory.Copula.claytonGeneratorDeriv θ s = -θ⁻¹ * (1 + s) ^ (-θ⁻¹ - 1)

theorem ProbabilityTheory.Copula.isC1_claytonGenerator (θ : ℝ) (hθ : 0 < θ) : (claytonGenerator θ hθ).IsC1 (claytonGeneratorDeriv θ)

Clayton's inverse generator is continuously differentiable on (0, ∞).

theorem ProbabilityTheory.Copula.kendallTau_clayton (θ : ℝ) (hθ : 0 < θ) : (clayton 2 θ hθ).kendallTau = θ / (θ + 2)

Nelsen, Section 5.1.1: Kendall's tau of the Clayton copula is θ / (θ + 2).

noncomputable def ProbabilityTheory.Copula.gumbelGeneratorDeriv (θ s : ℝ) : ℝ

The derivative of Gumbel's inverse generator exp(-s^(1/θ)).

Equations

• ProbabilityTheory.Copula.gumbelGeneratorDeriv θ s = Real.exp (-s ^ θ⁻¹) * -(θ⁻¹ * s ^ (θ⁻¹ - 1))

theorem ProbabilityTheory.Copula.isC1_gumbelGenerator (θ : ℝ) (hθ : 1 ≤ θ) : (gumbelGenerator θ hθ).IsC1 (gumbelGeneratorDeriv θ)

Gumbel's inverse generator is continuously differentiable on (0, ∞).

theorem ProbabilityTheory.Copula.kendallTau_gumbel (θ : ℝ) (hθ : 1 ≤ θ) : (gumbel θ hθ).kendallTau = 1 - 1 / θ

Nelsen, Section 5.1.1: Kendall's tau of the Gumbel–Hougaard copula is 1 − 1/θ.

theorem ProbabilityTheory.Copula.kendallTau_bb1 (θ : ℝ) (hθ : 0 < θ) (δ : ℝ) (hδ : 1 ≤ δ) : (bb1 θ hθ δ hδ).kendallTau = 1 - 2 / (δ * (θ + 2))

Kendall's tau of BB1: τ = 1 − 2 / (δ (θ + 2)).

theorem ProbabilityTheory.Copula.kendallTau_nelsen12 (θ : ℝ) (hθ : 1 ≤ θ) : (nelsen12 θ hθ).kendallTau = 1 - 2 / (3 * θ)

Kendall's tau of Nelsen's family 4.2.12: τ = 1 − 2/(3θ).

theorem ProbabilityTheory.Copula.kendallTau_nelsen14 (θ : ℝ) (hθ : 1 ≤ θ) : (nelsen14 θ hθ).kendallTau = (2 * θ - 1) / (2 * θ + 1)

Kendall's tau of Nelsen's family 4.2.14: τ = (2θ − 1)/(2θ + 1).

theorem ProbabilityTheory.Copula.isC1_truncatedLinearGenerator : truncatedLinearGenerator.IsC1 fun (x : ℝ) => -1

The non-strict generator ψ(s) = max(0, 1 − s) of W is C¹ on (0, 1), where it is positive.

theorem ProbabilityTheory.Copula.kendallTau_nelsen2 (θ : ℝ) (hθ : 1 ≤ θ) : (nelsen2 θ hθ).kendallTau = 1 - 2 / θ

Kendall's tau of Nelsen's family 4.2.2: τ = 1 − 2/θ.