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Tail coefficients of Archimax copulas under regular variation

Copula.ExtremeValue.Archimax expresses the tail coefficients of the Archimax copula C_{ψ,A}(u,v) = ψ(ℓ_A(φ(u), φ(v))) through limits of ratios of the inverse generator ψ (κ = 2A(1/2)):

λ_U = 2 - lim_{x→0+} (1 - ψ(κx))/(1 - ψ(x)), λ_L = lim_{x→∞} ψ(κx)/ψ(x) (strict generators).

Here we evaluate these limits under the regular-variation hypotheses of Capéraà, Fougères and Genest (2000, Section 4), stated on the generator φ:

• if φ(1 - at)/φ(1 - t) → a^m as t → 0+ for every a > 0 (φ(1 - 1/x) regularly varying with index -m at ∞, m > 0), then λ_U = 2 - (2A(1/2))^{1/m} (hasUpperTailDependence_archimaxCopula_of_regularVariation);
• if g is strict and φ(at)/φ(t) → a^{-k} as t → 0+ for every a > 0 (φ regularly varying at 0 with index -k, k > 0), then λ_L = (2A(1/2))^{-1/k} (hasLowerTailDependence_archimaxCopula_of_regularVariation);
• a non-strict generator gives λ_L = 0 as soon as A(1/2) > 1/2 (hasLowerTailDependence_archimaxCopula_zero_of_not_isStrict); the excluded case A(1/2) = 1/2 forces A(t) = max(t, 1-t) and C = M.

The passage from φ to ψ = φ^{-1} is the classical inversion of regular variation for monotone functions (tendsto_inverse_ratio_of_regularVariation, proved by a sandwich argument, no uniform convergence theorem needed). The hypotheses directly on ψ are hasUpperTailDependence_archimaxCopula_of_toFun_ratio and hasLowerTailDependence_archimaxCopula_of_toFun_ratio.

Examples: Gumbel's generator φ(t) = (-log t)^θ has m = θ; for it we verify the hypothesis and obtain λ_U = 2 - (2A(1/2))^{1/θ} for every Pickands function (hasUpperTailDependence_archimaxCopula_gumbelGenerator; A ≡ 1 recovers Gumbel's 2 - 2^{1/θ}). Clayton's generator φ(t) = (t^{-θ} - 1)/θ has k = θ, so A ≡ 1 gives the classical λ_L = 2^{-1/θ}.

References: P. Capéraà, A.-L. Fougères and C. Genest, Bivariate distributions with given extreme value attractor, J. Multivariate Anal. 72 (2000) 30–49, Section 4; N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation (1987), Theorem 1.5.12 (inverses).

theorem ProbabilityTheory.Copula.tendsto_const_mul_nhdsGT_zero {a : ℝ} (ha : 0 < a) : Filter.Tendsto (fun (x : ℝ) => a * x) (nhdsWithin 0 (Set.Ioi 0)) (nhdsWithin 0 (Set.Ioi 0))

theorem ProbabilityTheory.Copula.tendsto_inverse_ratio_of_regularVariation {F G : ℝ → ℝ} {m δ : ℝ} (hm : 0 < m) (hδ : 0 < δ) (hFmono : StrictMonoOn F (Set.Ioo 0 δ)) (hFpos : ∀ t ∈ Set.Ioo 0 δ, 0 < F t) (hF : ∀ (a : ℝ), 0 < a → Filter.Tendsto (fun (t : ℝ) => F (a * t) / F t) (nhdsWithin 0 (Set.Ioi 0)) (nhds (a ^ m))) (hG : Filter.Tendsto G (nhdsWithin 0 (Set.Ioi 0)) (nhdsWithin 0 (Set.Ioi 0))) (hFG : ∀ᶠ (y : ℝ) in nhdsWithin 0 (Set.Ioi 0), F (G y) = y) {s : ℝ} (hs : 0 < s) : Filter.Tendsto (fun (y : ℝ) => G (s * y) / G y) (nhdsWithin 0 (Set.Ioi 0)) (nhds (s ^ (1 / m)))

Inversion of regular variation at 0+. Let F be strictly increasing and positive on (0, δ) with F(at)/F(t) → a^m (t → 0+) for every a > 0, and let G(y) → 0+ be a right inverse of F near 0 (F(G(y)) = y). Then G(sy)/G(y) → s^{1/m} for every s > 0.

theorem ProbabilityTheory.Copula.BivariateGenerator.tendsto_toFun_zero (g : BivariateGenerator) : Filter.Tendsto g.toFun (nhdsWithin 0 (Set.Ioi 0)) (nhds 1)

ψ(x) → 1 as x → 0+.

theorem ProbabilityTheory.Copula.BivariateGenerator.tendsto_toFun_atTop (g : BivariateGenerator) : Filter.Tendsto g.toFun Filter.atTop (nhds 0)

For a strict generator, ψ(x) → 0 as x → ∞.

theorem ProbabilityTheory.Copula.BivariateGenerator.invFunReal_toFun (g : BivariateGenerator) {x : ℝ} (hx : 0 ≤ x) (hpos : 0 < g.toFun x) : g.invFunReal (g.toFun x) = x

φ(ψ(x)) = x whenever ψ(x) > 0, in terms of invFunReal.

theorem ProbabilityTheory.Copula.BivariateGenerator.tendsto_one_sub_toFun_ratio (g : BivariateGenerator) {m : ℝ} (hm : 0 < m) (hφ : ∀ (a : ℝ), 0 < a → Filter.Tendsto (fun (t : ℝ) => g.invFunReal (1 - a * t) / g.invFunReal (1 - t)) (nhdsWithin 0 (Set.Ioi 0)) (nhds (a ^ m))) {s : ℝ} (hs : 0 < s) : Filter.Tendsto (fun (x : ℝ) => (1 - g.toFun (s * x)) / (1 - g.toFun x)) (nhdsWithin 0 (Set.Ioi 0)) (nhds (s ^ (1 / m)))

Upper tail inversion: if φ(1 - at)/φ(1 - t) → a^m as t → 0+ for all a > 0, then (1 - ψ(sx))/(1 - ψ(x)) → s^{1/m} as x → 0+ for all s > 0.

theorem ProbabilityTheory.Copula.BivariateGenerator.IsStrict.tendsto_toFun_ratio_atTop {g : BivariateGenerator} (hg : g.IsStrict) {k : ℝ} (hk : 0 < k) (hφ : ∀ (a : ℝ), 0 < a → Filter.Tendsto (fun (t : ℝ) => g.invFunReal (a * t) / g.invFunReal t) (nhdsWithin 0 (Set.Ioi 0)) (nhds (a ^ (-k)))) {s : ℝ} (hs : 0 < s) : Filter.Tendsto (fun (x : ℝ) => g.toFun (s * x) / g.toFun x) Filter.atTop (nhds (s ^ (-1 / k)))

Lower tail inversion: for a strict generator with φ(at)/φ(t) → a^{-k} as t → 0+ for all a > 0, ψ(sx)/ψ(x) → s^{-1/k} as x → ∞ for all s > 0.

Archimax tail coefficients

theorem ProbabilityTheory.Copula.BivariateGenerator.two_mul_half_pos {A : ℝ → ℝ} (hA : IsPickandsFunction A) : 0 < 2 * A (1 / 2)

theorem ProbabilityTheory.Copula.BivariateGenerator.hasUpperTailDependence_archimaxCopula_of_toFun_ratio (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 ((2 * A (1 / 2)) ^ (1 / m)))) : (g.archimaxCopula A hA).HasUpperTailDependence (2 - (2 * A (1 / 2)) ^ (1 / m))

λ_U = 2 - (2A(1/2))^{1/m} when (1 - ψ(sx))/(1 - ψ(x)) → s^{1/m} at s = 2A(1/2).

theorem ProbabilityTheory.Copula.BivariateGenerator.hasUpperTailDependence_archimaxCopula_of_regularVariation (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) {m : ℝ} (hm : 0 < m) (hφ : ∀ (a : ℝ), 0 < a → Filter.Tendsto (fun (t : ℝ) => g.invFunReal (1 - a * t) / g.invFunReal (1 - t)) (nhdsWithin 0 (Set.Ioi 0)) (nhds (a ^ m))) : (g.archimaxCopula A hA).HasUpperTailDependence (2 - (2 * A (1 / 2)) ^ (1 / m))

Capéraà–Fougères–Genest, upper tail: if φ(1 - at)/φ(1 - t) → a^m as t → 0+ for every a > 0 (m > 0), then the Archimax copula has λ_U = 2 - (2A(1/2))^{1/m}.

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

λ_L = (2A(1/2))^{-1/k} for a strict generator when ψ(sx)/ψ(x) → s^{-1/k} as x → ∞ at s = 2A(1/2).

theorem ProbabilityTheory.Copula.BivariateGenerator.hasLowerTailDependence_archimaxCopula_of_regularVariation (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) (hg : g.IsStrict) {k : ℝ} (hk : 0 < k) (hφ : ∀ (a : ℝ), 0 < a → Filter.Tendsto (fun (t : ℝ) => g.invFunReal (a * t) / g.invFunReal t) (nhdsWithin 0 (Set.Ioi 0)) (nhds (a ^ (-k)))) : (g.archimaxCopula A hA).HasLowerTailDependence ((2 * A (1 / 2)) ^ (-1 / k))

Capéraà–Fougères–Genest, lower tail: for a strict generator with φ(at)/φ(t) → a^{-k} as t → 0+ for every a > 0 (k > 0), the Archimax copula has λ_L = (2A(1/2))^{-1/k}.

theorem ProbabilityTheory.Copula.BivariateGenerator.hasLowerTailDependence_archimaxCopula_zero_of_not_isStrict (g : BivariateGenerator) {A : ℝ → ℝ} (hA : IsPickandsFunction A) (hg : ¬g.IsStrict) (hA2 : 1 / 2 < A (1 / 2)) : (g.archimaxCopula A hA).HasLowerTailDependence 0

A non-strict generator gives no lower tail dependence as soon as A(1/2) > 1/2: the diagonal ψ(2A(1/2) φ(t)) vanishes near 0.

Example: Gumbel's generator

theorem ProbabilityTheory.Copula.tendsto_one_sub_exp_neg_div : Filter.Tendsto (fun (y : ℝ) => (1 - Real.exp (-y)) / y) (nhdsWithin 0 (Set.Ioi 0)) (nhds 1)

(1 - e^{-y})/y → 1 as y → 0+.

theorem ProbabilityTheory.Copula.tendsto_gumbelGenerator_ratio (θ : ℝ) (hθ : 1 ≤ θ) {s : ℝ} (hs : 0 < s) : Filter.Tendsto (fun (x : ℝ) => (1 - (gumbelGenerator θ hθ).toFun (s * x)) / (1 - (gumbelGenerator θ hθ).toFun x)) (nhdsWithin 0 (Set.Ioi 0)) (nhds (s ^ (1 / θ)))

For Gumbel's generator ψ(x) = exp(-x^{1/θ}): (1 - ψ(sx))/(1 - ψ(x)) → s^{1/θ} as x → 0+.

theorem ProbabilityTheory.Copula.hasUpperTailDependence_archimaxCopula_gumbelGenerator {A : ℝ → ℝ} (hA : IsPickandsFunction A) (θ : ℝ) (hθ : 1 ≤ θ) : ((gumbelGenerator θ hθ).archimaxCopula A hA).HasUpperTailDependence (2 - (2 * A (1 / 2)) ^ (1 / θ))

Gumbel–Archimax copulas: with Gumbel's generator ψ(x) = exp(-x^{1/θ}) (θ ≥ 1) and any Pickands function A, the Archimax copula has λ_U = 2 - (2A(1/2))^{1/θ} (Capéraà–Fougères–Genest 2000; A ≡ 1 gives Gumbel's 2 - 2^{1/θ}).