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Strict concordance monotonicity of Kendall's tau under full support

For bivariate copulas C ≤ D (pointwise, LowerOrthantLE) the difference of Kendall's taus splits as (LowerOrthantLE.kendallTau_sub_eq)

τ(D) - τ(C) = 4 (∫ (D - C) dD + ∫ (D - C) dC),

using the symmetry ∫ C dD = ∫ D dC of the concordance integral (integral_cdf_swap; Nelsen 2006, Thm. 5.1.1/Cor. 5.1.2). Both integrands are nonnegative and continuous, so if one of the two copulas has full support (its measure charges every nonempty open set) and C ≠ D, then τ(C) < τ(D) (LowerOrthantLE.kendallTau_lt_of_isOpenPosMeasure_right, ..._left). No claim is made for arbitrary comparable copulas.

theorem ProbabilityTheory.Copula.kendallTau_sub_eq (C D : Copula 2) : D.kendallTau - C.kendallTau = 4 * (∫ (x : Fin 2 → ↑unitInterval), D.cdf x - C.cdf x ∂D.toMeasure + ∫ (x : Fin 2 → ↑unitInterval), D.cdf x - C.cdf x ∂C.toMeasure)

The difference of Kendall's taus as two concordance integrals of D - C.

theorem ProbabilityTheory.Copula.LowerOrthantLE.kendallTau_lt_of_isOpenPosMeasure_right {C D : Copula 2} (h : C.LowerOrthantLE D) (hne : C ≠ D) [D.toMeasure.IsOpenPosMeasure] : C.kendallTau < D.kendallTau

Strict monotonicity of Kendall's tau when the larger copula has full support.

theorem ProbabilityTheory.Copula.LowerOrthantLE.kendallTau_lt_of_isOpenPosMeasure_left {C D : Copula 2} (h : C.LowerOrthantLE D) (hne : C ≠ D) [C.toMeasure.IsOpenPosMeasure] : C.kendallTau < D.kendallTau

Strict monotonicity of Kendall's tau when the smaller copula has full support.

theorem ProbabilityTheory.Copula.LowerOrthantLE.kendallTau_eq_iff_of_isOpenPosMeasure_right {C D : Copula 2} (h : C.LowerOrthantLE D) [D.toMeasure.IsOpenPosMeasure] : C.kendallTau = D.kendallTau ↔ C = D

Under full support of the larger copula, equal Kendall's taus force equality.

theorem ProbabilityTheory.Copula.LowerOrthantLE.kendallTau_eq_iff_of_isOpenPosMeasure_left {C D : Copula 2} (h : C.LowerOrthantLE D) [C.toMeasure.IsOpenPosMeasure] : C.kendallTau = D.kendallTau ↔ C = D

Under full support of the smaller copula, equal Kendall's taus force equality.