← Copula mathematical handbook

Spearman's rho of the Frank family in Debye form

Nelsen, An Introduction to Copulas, second edition, Example 5.8 and Table 4.1 (family 4.2.5); Genest (1987): for Frank's copula with parameter θ ≠ 0, ρ_θ = 1 − (12/θ) (D₁(θ) − D₂(θ)), where D₁ = debyeOne and D₂ = debyeTwo are the Debye functions of orders one and two.

The proof does not differentiate in θ under the integral sign. Writing Frank's cdf as θ⁻¹ L θ (θ u) (θ v) (FrankRho.L), the double integral is evaluated in Copula.Archimedean.SpearmanRhoFrankCore by expressing L θ x y as min x y + ∫_{max x y}^θ ∂_s L s x y ds and exchanging the order of integration. The case θ < 0 follows from ρ(C^{σ₂}) = −ρ(C) and the reflection identities D₁(−x) = D₁(x) + x/2, D₂(−x) = D₂(x) + 2x/3.

theorem ProbabilityTheory.Copula.FrankRho.L_comm (θ x y : ℝ) : L θ x y = L θ y x

theorem ProbabilityTheory.Copula.FrankRho.frank_cdf_eq_L (θ : ℝ) (hθ : 0 < θ) (u v : ↑unitInterval) : (frank θ hθ).cdf ![u, v] = θ⁻¹ * L θ (θ * ↑u) (θ * ↑v)

Frank's cdf on the whole closed unit square, as θ⁻¹ L θ (θ u) (θ v).

theorem ProbabilityTheory.Copula.spearmanRho_frank_integral (θ : ℝ) (hθ : 0 < θ) : (frank θ hθ).spearmanRho = 12 * (θ⁻¹ ^ 3 * ∫ (x : ℝ) (y : ℝ) in 0..θ, FrankRho.L θ x y) - 3

Double-integral form of Spearman's rho of Frank's copula (θ > 0): ρ = 12 θ⁻³ ∫₀^θ ∫₀^θ (-log (1 - (1 - e^{-x})(1 - e^{-y}) / (1 - e^{-θ}))) dy dx - 3.

theorem ProbabilityTheory.Copula.spearmanRho_frank_debye (θ : ℝ) (hθ : 0 < θ) : (frank θ hθ).spearmanRho = 1 - 12 / θ * (debyeOne θ - debyeTwo θ)

Spearman's rho of Frank's copula (Nelsen, Example 5.8; Genest 1987), θ > 0: ρ = 1 − (12/θ)(D₁(θ) − D₂(θ)).

theorem ProbabilityTheory.Copula.spearmanRho_frankNegative_debye (θ : ℝ) (hθ : θ < 0) : (frankNegative θ hθ).spearmanRho = 1 - 12 / θ * (debyeOne θ - debyeTwo θ)

Spearman's rho of Frank's copula for negative parameters, θ < 0: ρ = 1 − (12/θ)(D₁(θ) − D₂(θ)).