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The Schweizer–Wolff measure of dependence

The Schweizer–Wolff measure is σ(C) = 12 ∫∫ |C(u,v) - u v| du dv (Nelsen, An Introduction to Copulas, 2nd ed., Section 5.3). We prove: σ ≥ 0, σ(Π) = 0, σ(M) = σ(W) = 1, |ρ| ≤ σ, invariance under transposition and survival copulas, σ = |ρ| for quadrant dependent copulas (Nelsen Section 5.3), σ(C) = 0 if and only if C = Π, and the value |θ| / 3 on the Farlie–Gumbel–Morgenstern family.

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

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

The Schweizer–Wolff measure of dependence of a bivariate copula: σ(C) = 12 ∫∫ |C(u,v) - u v| du dv.

Equations

• C.schweizerWolff = 12 * ∫ (x : Fin 2 → ↑unitInterval), |C.cdf x - ↑(x 0) * ↑(x 1)| ∂(ProbabilityTheory.Copula.independence 2).toMeasure

theorem ProbabilityTheory.Copula.schweizerWolff_nonneg (C : Copula 2) : 0 ≤ C.schweizerWolff

theorem ProbabilityTheory.Copula.abs_spearmanRho_le_schweizerWolff (C : Copula 2) : |C.spearmanRho| ≤ C.schweizerWolff

Spearman's rho is bounded in absolute value by the Schweizer–Wolff measure.

@[simp]

theorem ProbabilityTheory.Copula.schweizerWolff_independence : (independence 2).schweizerWolff = 0

@[simp]

theorem ProbabilityTheory.Copula.schweizerWolff_transpose (C : Copula 2) : C.transpose.schweizerWolff = C.schweizerWolff

@[simp]

theorem ProbabilityTheory.Copula.schweizerWolff_survivalCopula (C : Copula 2) : C.survivalCopula.schweizerWolff = C.schweizerWolff

theorem ProbabilityTheory.Copula.eq_independence_of_schweizerWolff_eq_zero {C : Copula 2} (h : C.schweizerWolff = 0) : C = independence 2

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

theorem ProbabilityTheory.Copula.schweizerWolff_eq_zero_iff (C : Copula 2) : C.schweizerWolff = 0 ↔ C = independence 2

The Schweizer–Wolff measure vanishes exactly at the independence copula.

theorem ProbabilityTheory.Copula.IsPQD.mul_le_cdf {C : Copula 2} (h : C.IsPQD) (x : Fin 2 → ↑unitInterval) : ↑(x 0) * ↑(x 1) ≤ C.cdf x

theorem ProbabilityTheory.Copula.IsNQD.cdf_le_mul {C : Copula 2} (h : C.IsNQD) (x : Fin 2 → ↑unitInterval) : C.cdf x ≤ ↑(x 0) * ↑(x 1)

theorem ProbabilityTheory.Copula.IsPQD.schweizerWolff_eq_spearmanRho {C : Copula 2} (h : C.IsPQD) : C.schweizerWolff = C.spearmanRho

For positive quadrant dependent copulas, σ = ρ.

theorem ProbabilityTheory.Copula.IsNQD.schweizerWolff_eq_neg_spearmanRho {C : Copula 2} (h : C.IsNQD) : C.schweizerWolff = -C.spearmanRho

For negative quadrant dependent copulas, σ = -ρ.

theorem ProbabilityTheory.Copula.IsPQD.schweizerWolff_eq_abs_spearmanRho {C : Copula 2} (h : C.IsPQD) : C.schweizerWolff = |C.spearmanRho|

For positive quadrant dependent copulas, σ = |ρ|.

theorem ProbabilityTheory.Copula.IsNQD.schweizerWolff_eq_abs_spearmanRho {C : Copula 2} (h : C.IsNQD) : C.schweizerWolff = |C.spearmanRho|

For negative quadrant dependent copulas, σ = |ρ|.

@[simp]

theorem ProbabilityTheory.Copula.schweizerWolff_comonotonic : (comonotonic 2).schweizerWolff = 1

@[simp]

theorem ProbabilityTheory.Copula.schweizerWolff_countermonotonic : countermonotonic.schweizerWolff = 1

theorem ProbabilityTheory.Copula.schweizerWolff_fgm (θ : ℝ) (hθ : |θ| ≤ 1) : (fgm θ hθ).schweizerWolff = |θ| / 3

The Schweizer–Wolff measure of the Farlie–Gumbel–Morgenstern copula is |θ| / 3.