← Copula mathematical handbook

Kendall's tau of FGM copulas

The polynomial density gives the exact formula 2θ/9 on the whole admissible interval, as in Ansari–Rockel, Table 6.

theorem ProbabilityTheory.Copula.integral_unit_mul_one_sub_two_mul : ∫ (t : ↑unitInterval), ↑t * (1 - 2 * ↑t) = -1 / 6

theorem ProbabilityTheory.Copula.integral_unit_mul_one_sub_mul_one_sub_two_mul : ∫ (t : ↑unitInterval), ↑t * (1 - ↑t) * (1 - 2 * ↑t) = 0

theorem ProbabilityTheory.Copula.integral_fgm (θ : ℝ) (hθ : |θ| ≤ 1) (f : (Fin 2 → ↑unitInterval) → ℝ) : ∫ (x : Fin 2 → ↑unitInterval), f x ∂(fgm θ hθ).toMeasure = ∫ (x : Fin 2 → ↑unitInterval), fgmDensity θ x * f x

theorem ProbabilityTheory.Copula.kendallTau_fgm (θ : ℝ) (hθ : |θ| ≤ 1) : (fgm θ hθ).kendallTau = 2 * θ / 9