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Nelsen 2 dependence classification in Ansari–Rockel 2024

theorem Papers.AnsariRockel2024.nelsen2_not_pqd (θ : ℝ) (hθ : 1 ≤ θ) : ¬(ProbabilityTheory.Copula.nelsen2 θ hθ).IsPQD

Table 3: Nelsen 2 is not PQD at any finite admissible parameter.

theorem Papers.AnsariRockel2024.nelsen2_not_ci (θ : ℝ) (hθ : 1 ≤ θ) : ¬(ProbabilityTheory.Copula.nelsen2 θ hθ).IsCI

Table 3: no Nelsen 2 member is conditionally increasing.

theorem Papers.AnsariRockel2024.nelsen2_not_cd (θ : ℝ) (hθ : 1 < θ) : ¬(ProbabilityTheory.Copula.nelsen2 θ ⋯).IsCD

Table 3: parameters strictly above one fail conditional decrease.

theorem Papers.AnsariRockel2024.nelsen2_cd_one : (ProbabilityTheory.Copula.nelsen2 1 ⋯).IsCD

Table 3: the θ=1 lower-Fréchet endpoint is conditionally decreasing.

theorem Papers.AnsariRockel2024.nelsen2_cd_iff (θ : ℝ) (hθ : 1 ≤ θ) : (ProbabilityTheory.Copula.nelsen2 θ hθ).IsCD ↔ θ = 1

Table 3: the exact conditional-decrease range.

theorem Papers.AnsariRockel2024.nelsen2_not_tp2_cdf (θ : ℝ) (hθ : 1 ≤ θ) : ¬(ProbabilityTheory.Copula.nelsen2 θ hθ).IsTP2CDF

Table 3: the Nelsen 2 CDF is never TP2.

theorem Papers.AnsariRockel2024.nelsen2_not_mtp2_density (θ : ℝ) (hθ : 1 ≤ θ) : ¬(ProbabilityTheory.Copula.nelsen2 θ hθ).HasMTP2Density

Table 3: no Nelsen 2 member admits an MTP2 Lebesgue density.

theorem Papers.AnsariRockel2024.nelsen2_lowerOrthant {θ η : ℝ} (hθ : 1 ≤ θ) (hη : 1 ≤ η) (hθη : θ ≤ η) : (ProbabilityTheory.Copula.nelsen2 θ hθ).LowerOrthantLE (ProbabilityTheory.Copula.nelsen2 η hη)

Table 3: Nelsen 2 increases in lower-orthant order over its full parameter range.