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Khoudraji's asymmetrization

Khoudraji's device (A. Khoudraji, Contributions à l'étude des copules et à la modélisation des valeurs extrêmes bivariées, PhD thesis, Laval 1995; Genest, Ghoudi and Rivest 1998; E. Liebscher, Construction of asymmetric multivariate copulas, J. Multivariate Anal. 99 (2008)) turns a copula C and weights a, b ∈ [0,1] into

K_{a,b}(u,v) = u^{1-a} v^{1-b} C(u^a, v^b).

It is the special case D = Π of Liebscher's product construction C(u^a, v^b) D(u^{1-a}, v^{1-b}) (Copula.Transform.MaxProduct), which realizes it as the law of the coordinatewise maximum of a transformed sample of C and an independent transformed sample of Π. Results:

• the CDF formula (cdf_khoudraji), endpoint weights K_{1,1} = C, K_{0,0} = Π, and K_{a,b}(Π) = Π;
• Marshall–Olkin copulas are the Khoudraji asymmetrizations of M, and Tawn's asymmetric logistic copulas those of Gumbel's (khoudraji_comonotonic, tawn_eq_khoudraji);
• the construction preserves max-stability, the pointwise order, PQD and NQD, and commutes with transposition up to swapping the weights (transpose_khoudraji);
• it does break exchangeability: K_{a,b}(M) is exchangeable if and only if a = b or ab = 0 (isExchangeable_khoudraji_comonotonic_iff), detected by the directional Chatterjee's xi of the Marshall–Olkin copula.

noncomputable def ProbabilityTheory.Copula.khoudraji (C : Copula 2) (a b : ↑unitInterval) : Copula 2

Khoudraji's asymmetrization u^{1-a} v^{1-b} C(u^a, v^b) of a bivariate copula.

Equations

• C.khoudraji a b = C.maxProduct (ProbabilityTheory.Copula.independence 2) ![a, b]

theorem ProbabilityTheory.Copula.cdf_khoudraji (C : Copula 2) (a b u v : ↑unitInterval) : (C.khoudraji a b).cdf ![u, v] = C.cdf ![unitPower u ↑a ⋯, unitPower v ↑b ⋯] * (↑u ^ (1 - ↑a) * ↑v ^ (1 - ↑b))

The CDF of Khoudraji's asymmetrization: K(u,v) = C(u^a, v^b) u^{1-a} v^{1-b} (with 0^0 = 1).

@[simp]

theorem ProbabilityTheory.Copula.khoudraji_one_one (C : Copula 2) : C.khoudraji 1 1 = C

@[simp]

theorem ProbabilityTheory.Copula.khoudraji_zero_zero (C : Copula 2) : C.khoudraji 0 0 = independence 2

theorem ProbabilityTheory.Copula.unit_rpow_mul_rpow_one_sub (x a : ↑unitInterval) : ↑x ^ ↑a * ↑x ^ (1 - ↑a) = ↑x

x^a x^{1-a} = x on the unit interval (with 0^0 = 1).

@[simp]

theorem ProbabilityTheory.Copula.khoudraji_independence (a b : ↑unitInterval) : (independence 2).khoudraji a b = independence 2

@[simp]

theorem ProbabilityTheory.Copula.khoudraji_zero_left (C : Copula 2) (b : ↑unitInterval) : C.khoudraji 0 b = independence 2

A zero first weight gives independence: K_{0,b}(C) = Π.

@[simp]

theorem ProbabilityTheory.Copula.khoudraji_zero_right (C : Copula 2) (a : ↑unitInterval) : C.khoudraji a 0 = independence 2

A zero second weight gives independence: K_{a,0}(C) = Π.

theorem ProbabilityTheory.Copula.khoudraji_comonotonic (a b : ↑unitInterval) : (comonotonic 2).khoudraji a b = marshallOlkin a b

Marshall–Olkin copulas are the Khoudraji asymmetrizations of M.

theorem ProbabilityTheory.Copula.tawn_eq_khoudraji (θ : ℝ) (hθ : 1 ≤ θ) (α β : ↑unitInterval) : tawn θ hθ α β = (gumbel θ hθ).khoudraji α β

Tawn's asymmetric logistic copulas are the Khoudraji asymmetrizations of Gumbel's.

theorem ProbabilityTheory.Copula.IsExtremeValue.khoudraji {C : Copula 2} (hC : C.IsExtremeValue) (a b : ↑unitInterval) : (C.khoudraji a b).IsExtremeValue

Khoudraji's construction preserves max-stability (extreme-value copulas).

theorem ProbabilityTheory.Copula.transpose_khoudraji (C : Copula 2) (a b : ↑unitInterval) : (C.khoudraji a b).transpose = C.transpose.khoudraji b a

K_{a,b}(C)ᵀ = K_{b,a}(Cᵀ).

theorem ProbabilityTheory.Copula.LowerOrthantLE.khoudraji {C D : Copula 2} (h : C.LowerOrthantLE D) (a b : ↑unitInterval) : (C.khoudraji a b).LowerOrthantLE (D.khoudraji a b)

Khoudraji's construction is monotone for the pointwise order.

theorem ProbabilityTheory.Copula.IsPQD.khoudraji {C : Copula 2} (hC : C.IsPQD) (a b : ↑unitInterval) : (C.khoudraji a b).IsPQD

Khoudraji's construction preserves positive quadrant dependence.

theorem ProbabilityTheory.Copula.IsNQD.khoudraji {C : Copula 2} (hC : C.IsNQD) (a b : ↑unitInterval) : (C.khoudraji a b).IsNQD

Khoudraji's construction preserves negative quadrant dependence.

theorem ProbabilityTheory.Copula.IsExchangeable.khoudraji {C : Copula 2} (hC : C.IsExchangeable) (a : ↑unitInterval) : (C.khoudraji a a).IsExchangeable

Khoudraji's construction preserves exchangeability when the weights are equal.

theorem ProbabilityTheory.Copula.isExchangeable_khoudraji_comonotonic_iff (a b : ↑unitInterval) : ((comonotonic 2).khoudraji a b).IsExchangeable ↔ a = b ∨ a = 0 ∨ b = 0

The Khoudraji asymmetrization of M (the Marshall–Olkin copula) is exchangeable if and only if a = b or one weight vanishes (in which case it is Π).