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Concordance ordering of Archimedean copulas via generators

Nelsen, An Introduction to Copulas, second edition, Theorem 4.4.2: for Archimedean copulas C₁, C₂ with generators φ₁, φ₂, C₁ ≺ C₂ (pointwise C₁ ≤ C₂) iff f = φ₁ ∘ φ₂^[-1] is subadditive. We prove this for a strict second generator (ψ₂ > 0, so f = φ₁ ∘ ψ₂ is defined on all of [0, ∞)), with the first generator arbitrary (strict or not):

• BivariateGenerator.lowerOrthantLE_iff_subadditive: Theorem 4.4.2;
• subadditive_of_concaveOn and BivariateGenerator.lowerOrthantLE_of_concaveOn: Nelsen's Corollary 4.4.3 (a concave f with f(0) ≥ 0 is subadditive);
• BivariateGenerator.isPQD_iff (Nelsen, Section 4.4 / Exercise 4.19-type statement): a strict Archimedean copula is positively quadrant dependent iff ψ(x) ψ(y) ≤ ψ(x + y) (−log ψ subadditive); BivariateGenerator.not_isPQD_of_not_isStrict: non-strict Archimedean copulas are never PQD;
• BivariateGenerator.isNQD_iff: an Archimedean copula is negatively quadrant dependent iff φ(uv) ≤ φ(u) + φ(v) on (0, 1];
• lowerOrthantLE_clayton: Clayton's family is increasing in θ > 0 in the concordance order (f(s) = (1 + s)^{θ₁/θ₂} − 1 is concave for θ₁ ≤ θ₂). (For Gumbel's family the ordering is lowerOrthantLE_gumbel in Copula.Dependence.Gumbel.)

theorem ProbabilityTheory.Copula.subadditive_of_concaveOn {f : ℝ → ℝ} (hf : ConcaveOn ℝ (Set.Ici 0) f) (h0 : 0 ≤ f 0) {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) : f (x + y) ≤ f x + f y

Nelsen, Corollary 4.4.3: a concave function on [0, ∞) with f(0) ≥ 0 is subadditive.

theorem ProbabilityTheory.Copula.BivariateGenerator.lowerOrthantLE_iff_subadditive (g₁ g₂ : BivariateGenerator) (h₂ : g₂.IsStrict) : g₁.copula.LowerOrthantLE g₂.copula ↔ ∀ (x y : ℝ), 0 ≤ x → 0 ≤ y → g₁.invFunReal (g₂.toFun (x + y)) ≤ g₁.invFunReal (g₂.toFun x) + g₁.invFunReal (g₂.toFun y)

Nelsen, Theorem 4.4.2 (strict second generator): C₁ ≤ C₂ pointwise iff f = φ₁ ∘ ψ₂ is subadditive on [0, ∞).

theorem ProbabilityTheory.Copula.BivariateGenerator.lowerOrthantLE_of_concaveOn (g₁ g₂ : BivariateGenerator) (h₂ : g₂.IsStrict) (hf : ConcaveOn ℝ (Set.Ici 0) fun (s : ℝ) => g₁.invFunReal (g₂.toFun s)) : g₁.copula.LowerOrthantLE g₂.copula

Nelsen, Corollary 4.4.3: if f = φ₁ ∘ ψ₂ is concave on [0, ∞) (and ψ₂ is strict), then C₁ ≤ C₂.

theorem ProbabilityTheory.Copula.BivariateGenerator.isPQD_iff (g : BivariateGenerator) (hg : g.IsStrict) : g.copula.IsPQD ↔ ∀ (x y : ℝ), 0 ≤ x → 0 ≤ y → g.toFun x * g.toFun y ≤ g.toFun (x + y)

A strict Archimedean copula is PQD iff ψ(x) ψ(y) ≤ ψ(x + y) for all x, y ≥ 0 (equivalently, −log ψ is subadditive; Nelsen, Section 4.4).

theorem ProbabilityTheory.Copula.BivariateGenerator.not_isPQD_of_not_isStrict (g : BivariateGenerator) (hg : ¬g.IsStrict) : ¬g.copula.IsPQD

A non-strict Archimedean copula is never PQD: its diagonal vanishes near zero.

theorem ProbabilityTheory.Copula.BivariateGenerator.isNQD_iff (g : BivariateGenerator) : g.copula.IsNQD ↔ ∀ (u v : ℝ), 0 < u → u ≤ 1 → 0 < v → v ≤ 1 → g.invFunReal (u * v) ≤ g.invFunReal u + g.invFunReal v

An Archimedean copula is NQD iff φ(uv) ≤ φ(u) + φ(v) for all u, v ∈ (0, 1] (Nelsen, Section 4.4, with C₂ = Π in Theorem 4.4.2).

theorem ProbabilityTheory.Copula.lowerOrthantLE_clayton {θ η : ℝ} (hθ : 0 < θ) (hη : 0 < η) (hθη : θ ≤ η) : (clayton 2 θ hθ).LowerOrthantLE (clayton 2 η hη)

Clayton's family is increasing in θ > 0 in the lower-orthant (concordance) order: f(s) = φ_θ(ψ_η(s)) = (1 + s)^{θ/η} − 1 is concave for θ ≤ η (Nelsen, Example 4.19-type).