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Spearman's rho and Blomqvist's beta of an Archimedean copula in terms of its generator

For a bivariate Archimedean generator g with inverse generator ψ = g.toFun and generator φ = g.invFun, the CDF is C(u, v) = ψ(φ(u) + φ(v)) on the open square. Hence

• β = 4 ψ(2 φ(1/2)) - 1 (BivariateGenerator.blomqvistBeta_copula), and
• ρ = 12 ∫₀¹ ∫₀¹ C(u, v) du dv - 3 with C = g.cdf (BivariateGenerator.spearmanRho_copula), i.e. ρ = 12 ∫∫ ψ(φ(u) + φ(v)) du dv - 3.

For the individual table families the double integral collapses to a series, a one-dimensional integral or a closed form in Copula.Archimedean.SpearmanRhoAMH, SpearmanRhoNelsen9 and SpearmanRhoNelsen2; for most families (Clayton, Gumbel, Joe, ...) it has no elementary evaluation and this generic double-integral form is the statement.

theorem ProbabilityTheory.Copula.BivariateGenerator.blomqvistBeta_copula (g : BivariateGenerator) : g.copula.blomqvistBeta = 4 * g.toFun (2 * g.invFun unitHalf) - 1

Blomqvist's beta of an Archimedean copula: β = 4 ψ(2 φ(1/2)) - 1.

theorem ProbabilityTheory.Copula.BivariateGenerator.spearmanRho_copula (g : BivariateGenerator) : g.copula.spearmanRho = (12 * ∫ (v : ↑unitInterval) (u : ↑unitInterval), g.cdf u v) - 3

Spearman's rho of an Archimedean copula as a double integral of ψ(φ(u) + φ(v)).