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PQD/NQD classification of the one-parameter Archimedean families

One theorem per family of Nelsen, An Introduction to Copulas, Table 4.1 (numbers #1–#22), and for Gumbel's, Frank's, Joe's and the Ali–Mikhail–Haq family, stating positive and negative quadrant dependence on the full parameter range. Independence (Π) is both PQD and NQD; every other family below is at most one of the two. Notation: PQD is Copula.IsPQD, NQD is Copula.IsNQD.

#	family	range	PQD	NQD
1	Clayton	θ > 0	yes	no
1	Clayton	-1 ≤ θ < 0	no	yes
2		θ ≥ 1	no	iff θ = 1 (W)
3 / AMH		-1 ≤ θ ≤ 1	iff θ ≥ 0	iff θ ≤ 0
4 / Gumbel		θ ≥ 1	yes	iff θ = 1 (Π)
5 / Frank		θ > 0	yes	no
5 / Frank		θ < 0	no	yes
6 / Joe		θ ≥ 1	yes	iff θ = 1 (Π)
7		0 ≤ θ ≤ 1	iff θ = 1	yes
8		θ ≥ 1	no	iff θ ≤ 2
9, 10, 11		whole range	no	yes
12, 14		θ ≥ 1	yes	no
13		θ > 0	iff θ ≥ 1	iff θ ≤ 1
15		θ ≥ 1	no	iff θ = 1 (W)
16		θ ≥ 0	iff θ ≥ 1	iff θ = 0 (W)
17		θ ≠ 0	iff θ ≥ -1	iff θ ≤ -1
18		θ ≥ 2	no	no
19, 20		θ > 0	yes	no
21		θ ≥ 1	no	iff θ = 1 (W)
22		0 < θ ≤ 1	no	yes

Families 2, 8, 15, 16 (0 < θ < 1), 18, 21 (θ > 1) are neither PQD nor NQD on the indicated ranges: 2, 15, 21 (θ > 1), 18 have positive upper tail dependence, 16 lower tail dependence, 8 (θ > 2) exceeds Π at u = v = θ / (2 (θ - 1)).

theorem ProbabilityTheory.Copula.clayton_quadrant_pos (θ : ℝ) (hθ : 0 < θ) : (clayton 2 θ hθ).IsPQD ∧ ¬(clayton 2 θ hθ).IsNQD

#1 Clayton, θ > 0: PQD and not NQD.

theorem ProbabilityTheory.Copula.clayton_quadrant_neg (θ : ℝ) (hθ : -1 ≤ θ) (hn : θ < 0) : (claytonNegative θ hθ hn).IsNQD ∧ ¬(claytonNegative θ hθ hn).IsPQD

#1 Clayton, -1 ≤ θ < 0: NQD and not PQD.

theorem ProbabilityTheory.Copula.nelsen2_quadrant (θ : ℝ) (hθ : 1 ≤ θ) : ¬(nelsen2 θ hθ).IsPQD ∧ ((nelsen2 θ hθ).IsNQD ↔ θ = 1)

#2: never PQD; NQD iff θ = 1.

theorem ProbabilityTheory.Copula.amh_quadrant (θ : ℝ) (hmin : -1 ≤ θ) (hmax : θ ≤ 1) : ((amh θ hmin hmax).IsPQD ↔ 0 ≤ θ) ∧ ((amh θ hmin hmax).IsNQD ↔ θ ≤ 0)

#3 (Ali–Mikhail–Haq): PQD iff θ ≥ 0, NQD iff θ ≤ 0.

theorem ProbabilityTheory.Copula.gumbel_quadrant (θ : ℝ) (hθ : 1 ≤ θ) : (gumbel θ hθ).IsPQD ∧ ((gumbel θ hθ).IsNQD ↔ θ = 1)

#4 (Gumbel): always PQD; NQD iff θ = 1.

theorem ProbabilityTheory.Copula.frank_quadrant_pos (θ : ℝ) (hθ : 0 < θ) : (frank θ hθ).IsPQD ∧ ¬(frank θ hθ).IsNQD

#5 (Frank), θ > 0: PQD and not NQD.

theorem ProbabilityTheory.Copula.frank_quadrant_neg (θ : ℝ) (hθ : θ < 0) : (frankNegative θ hθ).IsNQD ∧ ¬(frankNegative θ hθ).IsPQD

#5 (Frank), θ < 0: NQD and not PQD.

theorem ProbabilityTheory.Copula.joe_quadrant (θ : ℝ) (hθ : 1 ≤ θ) : (joe θ hθ).IsPQD ∧ ((joe θ hθ).IsNQD ↔ θ = 1)

#6 (Joe): always PQD; NQD iff θ = 1.

theorem ProbabilityTheory.Copula.nelsen7_quadrant (θ : ↑unitInterval) : ((nelsen7 θ).IsPQD ↔ θ = 1) ∧ (nelsen7 θ).IsNQD

#7: always NQD; PQD iff θ = 1.

theorem ProbabilityTheory.Copula.nelsen8_quadrant (θ : ℝ) (hθ : 1 ≤ θ) : ¬(nelsen8 θ hθ).IsPQD ∧ ((nelsen8 θ hθ).IsNQD ↔ θ ≤ 2)

#8: never PQD; NQD iff θ ≤ 2.

theorem ProbabilityTheory.Copula.nelsen9_quadrant (θ : ℝ) (hθ : 0 < θ) (h1 : θ ≤ 1) : (nelsen9 θ hθ h1).IsNQD ∧ ¬(nelsen9 θ hθ h1).IsPQD

#9: NQD and not PQD.

theorem ProbabilityTheory.Copula.nelsen10_quadrant (θ : ℝ) (hθ : 0 < θ) (h1 : θ ≤ 1) : (nelsen10 θ hθ h1).IsNQD ∧ ¬(nelsen10 θ hθ h1).IsPQD

#10: NQD and not PQD.

theorem ProbabilityTheory.Copula.nelsen11_quadrant (θ : ℝ) (hθ : 0 < θ) (h2 : θ ≤ 1 / 2) : (nelsen11 θ hθ h2).IsNQD ∧ ¬(nelsen11 θ hθ h2).IsPQD

#11: NQD and not PQD.

theorem ProbabilityTheory.Copula.nelsen12_quadrant (θ : ℝ) (hθ : 1 ≤ θ) : (nelsen12 θ hθ).IsPQD ∧ ¬(nelsen12 θ hθ).IsNQD

#12: PQD and not NQD.

theorem ProbabilityTheory.Copula.nelsen13_quadrant (θ : ℝ) (hθ : 0 < θ) : ((nelsen13 θ hθ).IsPQD ↔ 1 ≤ θ) ∧ ((nelsen13 θ hθ).IsNQD ↔ θ ≤ 1)

#13: PQD iff θ ≥ 1, NQD iff θ ≤ 1.

theorem ProbabilityTheory.Copula.nelsen14_quadrant (θ : ℝ) (hθ : 1 ≤ θ) : (nelsen14 θ hθ).IsPQD ∧ ¬(nelsen14 θ hθ).IsNQD

#14: PQD and not NQD.

theorem ProbabilityTheory.Copula.nelsen15_quadrant (θ : ℝ) (hθ : 1 ≤ θ) : ¬(genestGhoudi θ hθ).IsPQD ∧ ((genestGhoudi θ hθ).IsNQD ↔ θ = 1)

#15 (Genest–Ghoudi): never PQD; NQD iff θ = 1.

theorem ProbabilityTheory.Copula.nelsen16_quadrant (θ : ℝ) (hθ : 0 ≤ θ) : ((nelsen16 θ hθ).IsPQD ↔ 1 ≤ θ) ∧ ((nelsen16 θ hθ).IsNQD ↔ θ = 0)

#16: PQD iff θ ≥ 1, NQD iff θ = 0.

theorem ProbabilityTheory.Copula.nelsen17_quadrant (θ : ℝ) (hθ : θ ≠ 0) : ((nelsen17 θ hθ).IsPQD ↔ -1 ≤ θ) ∧ ((nelsen17 θ hθ).IsNQD ↔ θ ≤ -1)

#17: PQD iff θ ≥ -1, NQD iff θ ≤ -1.

theorem ProbabilityTheory.Copula.nelsen18_quadrant (θ : ℝ) (hθ : 2 ≤ θ) : ¬(nelsen18 θ hθ).IsPQD ∧ ¬(nelsen18 θ hθ).IsNQD

#18: neither PQD nor NQD.

theorem ProbabilityTheory.Copula.nelsen19_quadrant (θ : ℝ) (hθ : 0 < θ) : (nelsen19 θ hθ).IsPQD ∧ ¬(nelsen19 θ hθ).IsNQD

#19: PQD and not NQD.

theorem ProbabilityTheory.Copula.nelsen20_quadrant (θ : ℝ) (hθ : 0 < θ) : (nelsen20 θ hθ).IsPQD ∧ ¬(nelsen20 θ hθ).IsNQD

#20: PQD and not NQD.

theorem ProbabilityTheory.Copula.nelsen21_quadrant (θ : ℝ) (hθ : 1 ≤ θ) : ¬(nelsen21 θ hθ).IsPQD ∧ ((nelsen21 θ hθ).IsNQD ↔ θ = 1)

#21: never PQD; NQD iff θ = 1.

theorem ProbabilityTheory.Copula.nelsen22_quadrant (θ : ℝ) (hθ : 0 < θ) (h1 : θ ≤ 1) : (nelsen22 θ hθ h1).IsNQD ∧ ¬(nelsen22 θ hθ h1).IsPQD

#22: NQD and not PQD.