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Table 5 / Appendix A.4.2: Schur order of the Fréchet and Mardia families

theorem Papers.AnsariRockel2024.mardia_schur_nonnegative {θ η : ℝ} (hθ : 0 ≤ θ) (hθη : θ ≤ η) (hη : η ≤ 1) : (ProbabilityTheory.Copula.mardia θ ⋯).SchurBothLE (ProbabilityTheory.Copula.mardia η ⋯)

Table 5 (* cell): Mardia increases in both-direction Schur order on 0≤θ≤1.

theorem Papers.AnsariRockel2024.mardia_schur_nonpositive {θ η : ℝ} (hθ : -1 ≤ θ) (hθη : θ ≤ η) (hη : η ≤ 0) : (ProbabilityTheory.Copula.mardia η ⋯).SchurBothLE (ProbabilityTheory.Copula.mardia θ ⋯)

Table 5 (* cell): Mardia decreases in both-direction Schur order on -1≤θ≤0.

theorem Papers.AnsariRockel2024.frechet_schur_mono_M {a a' : ℝ} (ha : 0 ≤ a) (haa : a ≤ a') (ha' : a' ≤ 1) : (ProbabilityTheory.Copula.frechet a 0 ha ⋯ ⋯).SchurBothLE (ProbabilityTheory.Copula.frechet a' 0 ⋯ ⋯ ⋯)

Table 5: on the unmixed axis β=0, Fréchet Schur order increases with the M weight.

theorem Papers.AnsariRockel2024.frechet_schur_mono_W {b b' : ℝ} (hb : 0 ≤ b) (hbb : b ≤ b') (hb' : b' ≤ 1) : (ProbabilityTheory.Copula.frechet 0 b ⋯ hb ⋯).SchurBothLE (ProbabilityTheory.Copula.frechet 0 b' ⋯ ⋯ ⋯)

Table 5: on the unmixed axis α=0, Fréchet Schur order increases with the W weight.