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Parameter ordering of Fréchet mixtures

Increasing the weight of M increases the copula; increasing the weight of W decreases it. This corrects the second direction printed in Ansari–Rockel, Table 5 and Appendix A.4.2.

theorem ProbabilityTheory.Copula.lowerOrthantLE_frechet {a b a' b' : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b ≤ 1) (ha' : 0 ≤ a') (hb' : 0 ≤ b') (hab' : a' + b' ≤ 1) (haa : a ≤ a') (hbb : b' ≤ b) : (frechet a b ha hb hab).LowerOrthantLE (frechet a' b' ha' hb' hab')

theorem ProbabilityTheory.Copula.not_lowerOrthantLE_independence_countermonotonic : ¬(independence 2).LowerOrthantLE countermonotonic

The reverse direction in the lower-bound weight already fails at the two parameter endpoints, as witnessed by the center of the square.