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Archimax copulas

Capéraà, Fougères and Genest (2000) combine an Archimedean generator with a Pickands dependence function. For a bivariate Archimedean generator g (inverse generator ψ = g.toFun, generator φ = g.invFun) and a Pickands function A, the Archimax copula is

C_{ψ,A}(u,v) = ψ(ℓ_A(φ(u), φ(v))) = ψ((φ(u) + φ(v)) A(φ(v) / (φ(u) + φ(v))))

on (0,1]² (zero on the lower edges), where ℓ_A = pickandsTail A is the stable tail dependence function. We use the library's convention for A (that of pickandsCopula, where C_A(u,v) = exp(-ℓ_A(-log u, -log v))); Capéraà–Fougères–Genest write A(φ(u)/(φ(u)+φ(v))), i.e. their A is our t ↦ A(1 - t) (archimaxCopula_comp_one_sub relates the two).

Main results:

• archimaxCopula g A hA: C_{ψ,A} is a copula (Capéraà–Fougères–Genest 2000, Proposition 1). The proof is elementary: ℓ_A is nondecreasing and submodular, and ψ is convex and nonincreasing, so ψ ∘ ℓ_A is supermodular (this is the argument behind exp ∘ (-ℓ_A) for the extreme-value case).
• archimaxCopula_const_one: A ≡ 1 gives the Archimedean copula g.copula; archimaxCopula_exponentialGenerator: ψ = exp(-·) gives the extreme-value copula pickandsCopula A; archimaxCopula_max: A = max(t, 1-t) gives M for every generator.
• archimaxCopula_lowerOrthantLE: A ≤ B implies C_{ψ,B} ≤ C_{ψ,A}; in particular g.copula ≤ C_{ψ,A} (generator_lowerOrthantLE_archimaxCopula).
• transpose_archimaxCopula: the transpose is the Archimax copula of t ↦ A(1 - t).
• diagonal_archimaxCopula: the diagonal is ψ(2A(1/2) φ(t)), hence Blomqvist's β = 4ψ(2A(1/2) φ(1/2)) - 1 (blomqvistBeta_archimaxCopula).
• Tail dependence (Capéraà–Fougères–Genest 2000, Section 4): if ψ has a finite nonzero right derivative at 0 then λ_U = 2(1 - A(1/2)) (hasUpperTailDependence_archimaxCopula_of_hasDerivWithinAt); in general λ_U = 2 - lim_{x→0+} (1 - ψ(2A(1/2)x)) / (1 - ψ(x)) (hasUpperTailDependence_archimaxCopula_of_tendsto), and for a strict generator λ_L = lim_{x→∞} ψ(2A(1/2)x) / ψ(x) (hasLowerTailDependence_archimaxCopula_of_tendsto).

References: P. Capéraà, A.-L. Fougères and C. Genest, Bivariate distributions with given extreme value attractor, J. Multivariate Anal. 72 (2000) 30–49; F. Durante and C. Sempi, Principles of Copula Theory (2016), Section 6.6 (Archimax copulas).

noncomputable def ProbabilityTheory.Copula.BivariateGenerator.archimaxCDF (g : BivariateGenerator) (A : ℝ → ℝ) (u v : ↑unitInterval) : ℝ

The Archimax CDF ψ(ℓ_A(φ(u), φ(v))), zero on the lower edges.

Equations

• g.archimaxCDF A u v = if u = 0 ∨ v = 0 then 0 else g.toFun (ProbabilityTheory.Copula.pickandsTail A (g.invFun u) (g.invFun v))

theorem ProbabilityTheory.Copula.BivariateGenerator.archimaxCDF_of_ne (g : BivariateGenerator) {A : ℝ → ℝ} {u v : ↑unitInterval} (hu : u ≠ 0) (hv : v ≠ 0) : g.archimaxCDF A u v = g.toFun (pickandsTail A (g.invFun u) (g.invFun v))

@[simp]

theorem ProbabilityTheory.Copula.BivariateGenerator.archimaxCDF_zero_left (g : BivariateGenerator) (A : ℝ → ℝ) (v : ↑unitInterval) : g.archimaxCDF A 0 v = 0

@[simp]

theorem ProbabilityTheory.Copula.BivariateGenerator.archimaxCDF_zero_right (g : BivariateGenerator) (A : ℝ → ℝ) (u : ↑unitInterval) : g.archimaxCDF A u 0 = 0

theorem ProbabilityTheory.Copula.BivariateGenerator.archimaxCDF_nonneg (g : BivariateGenerator) (A : ℝ → ℝ) (u v : ↑unitInterval) (hA : IsPickandsFunction A) : 0 ≤ g.archimaxCDF A u v

theorem ProbabilityTheory.Copula.BivariateGenerator.archimaxCDF_one_left (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) (v : ↑unitInterval) : g.archimaxCDF A 1 v = ↑v

theorem ProbabilityTheory.Copula.BivariateGenerator.archimaxCDF_one_right (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) (u : ↑unitInterval) : g.archimaxCDF A u 1 = ↑u

theorem ProbabilityTheory.Copula.BivariateGenerator.archimaxCDF_mono (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) {u u' v v' : ↑unitInterval} (hu : u ≤ u') (hv : v ≤ v') : g.archimaxCDF A u v ≤ g.archimaxCDF A u' v'

theorem ProbabilityTheory.Copula.BivariateGenerator.add_le_add_of_submodular (g : BivariateGenerator) {p q r s : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) (hpr : p ≤ r) (hps : p ≤ s) (hqrs : q ≤ r + s - p) : g.toFun r + g.toFun s ≤ g.toFun p + g.toFun q

Supermodularity of ψ ∘ ℓ: for a convex nonincreasing ψ on [0,∞), p ≤ r, p ≤ s, 0 ≤ q ≤ r + s - p imply ψ(r) + ψ(s) ≤ ψ(p) + ψ(q).

theorem ProbabilityTheory.Copula.BivariateGenerator.archimaxCDF_rect (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) {a b c e : ↑unitInterval} (hab : a ≤ b) (hce : c ≤ e) : 0 ≤ g.archimaxCDF A b e - g.archimaxCDF A a e - g.archimaxCDF A b c + g.archimaxCDF A a c

noncomputable def ProbabilityTheory.Copula.BivariateGenerator.archimaxCopula (g : BivariateGenerator) (A : ℝ → ℝ) (hA : IsPickandsFunction A) : Copula 2

The Archimax copula C_{ψ,A}(u,v) = ψ(ℓ_A(φ(u), φ(v))) of a bivariate Archimedean generator and a Pickands dependence function (Capéraà–Fougères–Genest 2000).

Equations

• g.archimaxCopula A hA = ProbabilityTheory.Copula.ofClassical (fun (u : Fin 2 → ↑unitInterval) => g.archimaxCDF A (u 0) (u 1)) ⋯

theorem ProbabilityTheory.Copula.BivariateGenerator.cdf_archimaxCopula (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) (u : Fin 2 → ↑unitInterval) : (g.archimaxCopula A hA).cdf u = g.archimaxCDF A (u 0) (u 1)

theorem ProbabilityTheory.Copula.BivariateGenerator.cdf_archimaxCopula_two (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) (u v : ↑unitInterval) : (g.archimaxCopula A hA).cdf ![u, v] = g.archimaxCDF A u v

theorem ProbabilityTheory.Copula.BivariateGenerator.cdf_archimaxCopula_eq (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) {u v : ↑unitInterval} (hu : u ≠ 0) (hv : v ≠ 0) : (g.archimaxCopula A hA).cdf ![u, v] = g.toFun ((g.invFun u + g.invFun v) * A (g.invFun v / (g.invFun u + g.invFun v)))

The Capéraà–Fougères–Genest formula C(u,v) = ψ((φ(u) + φ(v)) A(φ(v) / (φ(u) + φ(v)))) on (0,1]².

Special cases

theorem ProbabilityTheory.Copula.BivariateGenerator.archimaxCopula_const_one (g : BivariateGenerator) : g.archimaxCopula (fun (x : ℝ) => 1) isPickandsFunction_const_one = g.copula

A ≡ 1 gives the Archimedean copula of the generator.

theorem ProbabilityTheory.Copula.archimaxCopula_exponentialGenerator {A : ℝ → ℝ} (hA : IsPickandsFunction A) : exponentialGenerator.archimaxCopula A hA = pickandsCopula A hA

ψ = exp(-·) (φ = -log) gives the extreme-value copula C_A.

theorem ProbabilityTheory.Copula.BivariateGenerator.archimaxCopula_max (g : BivariateGenerator) : g.archimaxCopula (fun (t : ℝ) => max t (1 - t)) isPickandsFunction_max = comonotonic 2

A(t) = max(t, 1-t) gives the upper Fréchet bound M, whatever the generator.

Order

theorem ProbabilityTheory.Copula.BivariateGenerator.archimaxCopula_lowerOrthantLE (g : BivariateGenerator) {A B : ℝ → ℝ} (hA : IsPickandsFunction A) (hB : IsPickandsFunction B) (h : ∀ t ∈ Set.Icc 0 1, A t ≤ B t) : (g.archimaxCopula B hB).LowerOrthantLE (g.archimaxCopula A hA)

A pointwise larger Pickands function gives a pointwise smaller Archimax copula.

theorem ProbabilityTheory.Copula.BivariateGenerator.generator_lowerOrthantLE_archimaxCopula (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) : g.copula.LowerOrthantLE (g.archimaxCopula A hA)

Every Archimax copula dominates the Archimedean copula of its generator.

theorem ProbabilityTheory.Copula.BivariateGenerator.transpose_archimaxCopula (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) : (g.archimaxCopula A hA).transpose = g.archimaxCopula (fun (t : ℝ) => A (1 - t)) ⋯

The transpose of an Archimax copula is the Archimax copula of t ↦ A(1 - t).

theorem ProbabilityTheory.Copula.BivariateGenerator.cdf_archimaxCopula_comp_one_sub (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) {u v : ↑unitInterval} (hu : u ≠ 0) (hv : v ≠ 0) : (g.archimaxCopula (fun (t : ℝ) => A (1 - t)) ⋯).cdf ![u, v] = g.toFun ((g.invFun u + g.invFun v) * A (g.invFun u / (g.invFun u + g.invFun v)))

The Capéraà–Fougères–Genest convention: with A' (t) = A (1 - t), the copula is ψ((φ(u) + φ(v)) A'(φ(u) / (φ(u) + φ(v)))).

Diagonal, Blomqvist's beta and tail dependence

theorem ProbabilityTheory.Copula.BivariateGenerator.diagonal_archimaxCopula (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) {t : ↑unitInterval} (ht : t ≠ 0) : (g.archimaxCopula A hA).diagonal t = g.toFun (2 * A (1 / 2) * g.invFun t)

The diagonal section δ(t) = ψ(2A(1/2) φ(t)) for t ≠ 0.

theorem ProbabilityTheory.Copula.BivariateGenerator.blomqvistBeta_archimaxCopula (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) : (g.archimaxCopula A hA).blomqvistBeta = 4 * g.toFun (2 * A (1 / 2) * g.invFun unitHalf) - 1

Blomqvist's beta of an Archimax copula: β = 4ψ(2A(1/2) φ(1/2)) - 1.

theorem ProbabilityTheory.Copula.BivariateGenerator.hasDerivWithinAt_diagonal_archimaxCopula_one (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) {m : ℝ} (h : Filter.Tendsto (fun (x : ℝ) => (1 - g.toFun (2 * A (1 / 2) * x)) / (1 - g.toFun x)) (nhdsWithin 0 (Set.Ioi 0)) (nhds m)) : HasDerivWithinAt (fun (x : ℝ) => (g.archimaxCopula A hA).diagonal (Set.projIcc 0 1 ⋯ x)) m (Set.Icc 0 1) 1

The left derivative of the Archimax diagonal at one: if (1 - ψ(κx)) / (1 - ψ(x)) → m as x → 0+ with κ = 2A(1/2), then δ'(1⁻) = m.

theorem ProbabilityTheory.Copula.BivariateGenerator.hasUpperTailDependence_archimaxCopula_of_tendsto (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) {m : ℝ} (h : Filter.Tendsto (fun (x : ℝ) => (1 - g.toFun (2 * A (1 / 2) * x)) / (1 - g.toFun x)) (nhdsWithin 0 (Set.Ioi 0)) (nhds m)) : (g.archimaxCopula A hA).HasUpperTailDependence (2 - m)

Upper tail dependence of Archimax copulas (Capéraà–Fougères–Genest 2000): λ_U = 2 - lim_{x → 0+} (1 - ψ(2A(1/2) x)) / (1 - ψ(x)) whenever the limit exists.

theorem ProbabilityTheory.Copula.BivariateGenerator.tendsto_archimaxRatio_of_hasDerivWithinAt (g : BivariateGenerator) {d κ : ℝ} (hd : HasDerivWithinAt g.toFun d (Set.Ici 0) 0) (hd0 : d ≠ 0) (hκ : 0 < κ) : Filter.Tendsto (fun (x : ℝ) => (1 - g.toFun (κ * x)) / (1 - g.toFun x)) (nhdsWithin 0 (Set.Ioi 0)) (nhds κ)

If ψ has a finite nonzero right derivative at 0, then (1 - ψ(κx)) / (1 - ψ(x)) → κ as x → 0+ for every κ > 0.

theorem ProbabilityTheory.Copula.BivariateGenerator.hasUpperTailDependence_archimaxCopula_of_hasDerivWithinAt (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) {d : ℝ} (hd : HasDerivWithinAt g.toFun d (Set.Ici 0) 0) (hd0 : d ≠ 0) : (g.archimaxCopula A hA).HasUpperTailDependence (2 * (1 - A (1 / 2)))

Upper tail coefficient λ_U = 2(1 - A(1/2)) of an Archimax copula whose inverse generator has a finite nonzero right derivative at 0 (equivalently φ'(1⁻) ≠ 0; the Archimedean part then has no upper tail dependence and the tail behaviour is that of the extreme-value attractor C_A, Capéraà–Fougères–Genest 2000, Section 4).

theorem ProbabilityTheory.Copula.BivariateGenerator.lowerTailRatio_archimaxCopula (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) {t : ↑unitInterval} (ht : t ≠ 0) : (g.archimaxCopula A hA).lowerTailRatio t = g.toFun (2 * A (1 / 2) * g.invFun t) / g.toFun (g.invFun t)

The lower tail ratio along the generator: δ(t) / t = ψ(2A(1/2) φ(t)) / ψ(φ(t)).

theorem ProbabilityTheory.Copula.BivariateGenerator.hasLowerTailDependence_archimaxCopula_of_tendsto (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) (hg : g.IsStrict) {l : ℝ} (h : Filter.Tendsto (fun (x : ℝ) => g.toFun (2 * A (1 / 2) * x) / g.toFun x) Filter.atTop (nhds l)) : (g.archimaxCopula A hA).HasLowerTailDependence l

Lower tail dependence of Archimax copulas with a strict generator: λ_L = lim_{x → ∞} ψ(2A(1/2) x) / ψ(x) whenever this limit exists.