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Kendall's tau of families 21 and 22 of Nelsen's Table 4.1

Explicit integral forms of τ = 1 + 4 ∫₀¹ φ(t) / φ'(t) dt (Nelsen, Corollary 5.1.4, BivariateGenerator.kendallTau_eq_of_hasDerivAt) for the two non-strict families with non-elementary Kendall integral:

• family 4.2.21, θ ≥ 1, φ(t) = 1 − (1 − (1 − t)^θ)^{1/θ}, with w(t) = 1 − (1 − t)^θ: τ = 1 − 4 ∫₀¹ (1 − t)^{1−θ} w(t)^{1 − 1/θ} (1 − w(t)^{1/θ}) dt (kendallTau_nelsen21);
• family 4.2.22, 0 < θ ≤ 1, φ(t) = arcsin (1 − t^θ): τ = 1 − (4/θ) ∫₀¹ t^{1 − θ/2} √(2 − t^θ) arcsin (1 − t^θ) dt (kendallTau_nelsen22).

theorem ProbabilityTheory.Copula.kendallTau_nelsen21 (θ : ℝ) (hθ : 1 ≤ θ) : (nelsen21 θ hθ).kendallTau = 1 - 4 * ∫ (t : ℝ) in 0..1, (1 - t) ^ (1 - θ) * (1 - (1 - t) ^ θ) ^ (1 - θ⁻¹) * (1 - (1 - (1 - t) ^ θ) ^ θ⁻¹)

Nelsen, Table 4.1, family 21 (θ ≥ 1), with w(t) = 1 − (1 − t)^θ: τ = 1 − 4 ∫₀¹ (1 − t)^{1−θ} w(t)^{1 − 1/θ} (1 − w(t)^{1/θ}) dt.

theorem ProbabilityTheory.Copula.kendallTau_nelsen22 (θ : ℝ) (hθ : 0 < θ) (h1 : θ ≤ 1) : (nelsen22 θ hθ h1).kendallTau = 1 - 4 / θ * ∫ (t : ℝ) in 0..1, t ^ (1 - θ / 2) * √(2 - t ^ θ) * Real.arcsin (1 - t ^ θ)

Nelsen, Table 4.1, family 22 (0 < θ ≤ 1): τ = 1 − (4/θ) ∫₀¹ t^{1−θ/2} √(2 − t^θ) arcsin (1 − t^θ) dt.