Exact AMH Schur order on the two conditional-monotonicity regions #
theorem
Papers.AnsariRockel2024.amh_schur_nonnegative_iff
{θ η : ℝ}
(hθ : 0 ≤ θ)
(hθmax : θ ≤ 1)
(hη : 0 ≤ η)
(hηmax : η ≤ 1)
:
(ProbabilityTheory.Copula.amh θ ⋯ hθmax).SchurBothLE (ProbabilityTheory.Copula.amh η ⋯ hηmax) ↔ θ ≤ η
Table 3: on the nonnegative parameter region, AMH Schur order is parameter order.
theorem
Papers.AnsariRockel2024.amh_schur_nonpositive_iff
{θ η : ℝ}
(hθmin : -1 ≤ θ)
(hθ : θ ≤ 0)
(hηmin : -1 ≤ η)
(hη : η ≤ 0)
:
(ProbabilityTheory.Copula.amh θ hθmin ⋯).SchurBothLE (ProbabilityTheory.Copula.amh η hηmin ⋯) ↔ η ≤ θ
Table 3: on the nonpositive region, AMH Schur order reverses parameter order.