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Quadrant dependence of Ali–Mikhail–Haq copulas

theorem ProbabilityTheory.Copula.isPQD_amh (θ : ℝ) (hmin : -1 ≤ θ) (hmax : θ ≤ 1) (hθ : 0 ≤ θ) : (amh θ hmin hmax).IsPQD

Nonnegative AMH parameters are positively quadrant dependent.

theorem ProbabilityTheory.Copula.isNQD_amh (θ : ℝ) (hmin : -1 ≤ θ) (hmax : θ ≤ 1) (hθ : θ ≤ 0) : (amh θ hmin hmax).IsNQD

Nonpositive AMH parameters are negatively quadrant dependent.

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

AMH is PQD exactly for nonnegative parameters.

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

AMH is NQD exactly for nonpositive parameters.