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Hoeffding's dependence index

Hoeffding's measure is Φ²(C) = 90 ∫∫ (C(u,v) - u v)² du dv (Nelsen, An Introduction to Copulas, 2nd ed., Section 5.3). We prove Φ² ≥ 0, Φ²(Π) = 0, invariance under transposition and survival copulas, Φ²(C) = 0 if and only if C = Π, and Φ²(FGM θ) = θ² / 10.

The sharp upper bound Φ² ≤ 1, with equality exactly at M and W, is in Copula.Measures.Bounds.

noncomputable def ProbabilityTheory.Copula.hoeffdingPhiSq (C : Copula 2) : ℝ

Hoeffding's dependence index Φ²(C) = 90 ∫∫ (C(u,v) - u v)² du dv.

Equations

• C.hoeffdingPhiSq = 90 * ∫ (x : Fin 2 → ↑unitInterval), (C.cdf x - ↑(x 0) * ↑(x 1)) ^ 2 ∂(ProbabilityTheory.Copula.independence 2).toMeasure

theorem ProbabilityTheory.Copula.hoeffdingPhiSq_nonneg (C : Copula 2) : 0 ≤ C.hoeffdingPhiSq

@[simp]

theorem ProbabilityTheory.Copula.hoeffdingPhiSq_independence : (independence 2).hoeffdingPhiSq = 0

@[simp]

theorem ProbabilityTheory.Copula.hoeffdingPhiSq_transpose (C : Copula 2) : C.transpose.hoeffdingPhiSq = C.hoeffdingPhiSq

@[simp]

theorem ProbabilityTheory.Copula.hoeffdingPhiSq_survivalCopula (C : Copula 2) : C.survivalCopula.hoeffdingPhiSq = C.hoeffdingPhiSq

theorem ProbabilityTheory.Copula.eq_independence_of_hoeffdingPhiSq_eq_zero {C : Copula 2} (h : C.hoeffdingPhiSq = 0) : C = independence 2

Φ²(C) = 0 forces C to be the independence copula.

theorem ProbabilityTheory.Copula.hoeffdingPhiSq_eq_zero_iff (C : Copula 2) : C.hoeffdingPhiSq = 0 ↔ C = independence 2

Hoeffding's index vanishes exactly at the independence copula.

theorem ProbabilityTheory.Copula.hoeffdingPhiSq_fgm (θ : ℝ) (hθ : |θ| ≤ 1) : (fgm θ hθ).hoeffdingPhiSq = θ ^ 2 / 10

Hoeffding's index of the Farlie–Gumbel–Morgenstern copula is θ² / 10.