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Exact conditional monotonicity of Ali–Mikhail–Haq copulas

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

AMH is directionally conditionally increasing for every nonnegative parameter.

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

AMH is directionally conditionally decreasing for every nonpositive parameter.

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

AMH is conditionally increasing exactly for θ≥0.

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

AMH is conditionally decreasing exactly for θ≤0.