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Kendall's tau of the Ali–Mikhail–Haq family

For the Ali–Mikhail–Haq family (Nelsen, An Introduction to Copulas, second edition, family 4.2.3), -1 ≤ θ < 1, θ ≠ 0, τ = 1 − 2 (θ + (1 − θ)² log(1 − θ)) / (3 θ²) (kendallTau_amh; Nelsen, Section 5.1.1).

The inverse generator ψ(s) = (1 − θ) / (eˢ − θ) is strict and smooth (isC1_amhGenerator). With w = 1 − θ + θ t, the Kendall integrand is φ(t)/φ'(t) = −t w (log w − log t) / (1 − θ), which is integrated with an explicit antiderivative. For θ = 0 the family is the independence copula (amh_zero), with τ = 0.

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

The derivative of the Ali–Mikhail–Haq inverse generator (1 − θ) / (eˢ − θ).

Equations

• ProbabilityTheory.Copula.amhGeneratorDeriv θ s = -(1 - θ) * Real.exp s / (Real.exp s - θ) ^ 2

theorem ProbabilityTheory.Copula.isC1_amhGenerator (θ : ℝ) (hmin : -1 ≤ θ) (hmax : θ < 1) : (amhGenerator θ hmin hmax).IsC1 (amhGeneratorDeriv θ)

The Ali–Mikhail–Haq inverse generator is continuously differentiable on (0, ∞).

theorem ProbabilityTheory.Copula.kendallTau_amh (θ : ℝ) (hmin : -1 ≤ θ) (hmax : θ < 1) (hθ : θ ≠ 0) : (amh θ hmin ⋯).kendallTau = 1 - 2 * (θ + (1 - θ) ^ 2 * Real.log (1 - θ)) / (3 * θ ^ 2)

Kendall's tau of the Ali–Mikhail–Haq copula: τ = 1 − 2 (θ + (1 − θ)² log(1 − θ)) / (3 θ²) for -1 ≤ θ < 1, θ ≠ 0.