Further claims of the revised manuscript #
Checks for statements in exact-blest-regions.tex that sit outside the labeled results or were
added in the revision: the dual-certificate lemma for general continuous costs, the derivative formulas for Υ used in the corollary on (ν,νᵀ), the
symmetries (and the failure of central symmetry) of the two regions, the trivial bound 1/2,
the numerical fibres in the discussion, the non-extremality of A_w for w > 1/2, the shuffle
description of the ρ - η extremizers, and the values quoted in the figure captions.
The dual-certificate lemma #
Part (a) of the dual-certificate lemma: for any coupling of two uniform laws (the law of an
arbitrary copula), any continuous cost c, and continuous potentials with c ≤ φ ⊕ ψ on the
unit square, the expected cost is at most ∫ φ + ∫ ψ, with equality exactly when the coupling
is concentrated on the contact set.
Part (b) of the dual-certificate lemma: a coupling concentrated on the graph of a
measurable map over the first coordinate, up to finitely many vertical and horizontal lines,
is the law of (X, T(X)).
Part (b) with the roles of the coordinates exchanged.
Normalization and the two theorems in the form stated in the introduction #
The identity 2r - Φ(r) = -Φ(-r) stated after the definition of Φ.
In Theorem 1.1, the unique lower-boundary copula is the survival copula of the upper one.
In Theorem 1.2, the unique lower-boundary copula is the transpose of the upper one.
The fibre above η = e has length 2 Υ(e), the largest asymmetry at that η.
Symmetries of the two regions #
C ↦ C^⊥ negates (ρ,ν), so the (ρ,ν)-region is symmetric about the origin.
Survival reflects every (ρ,ν)-fibre about the diagonal.
Transposition reflects every (η,ν)-fibre about the diagonal.
The (η,ν)-region is not symmetric about the origin.
The bound 1/2 from the (ρ,ν)-region alone #
ν(C) - ρ(C) and ν(Cᵀ) - ρ(C) cannot be extreme in opposite directions.
Derivatives of Υ #
The inverse parameter e ↦ a of the randomized branch, as a real function.
Equations
Instances For
Dividing the two a-derivatives gives the slope (3a-1)/(1+a), as in the proof.
Numerical fibres in the discussion #
Remark on randomization and the by-product #
For w ∈ (1/2,1), A_w lies strictly below the upper boundary.
In Lemma 4.1, η(B_a) → -1 as a → 1.
The panels of the Spearman figure: c = 1/4, 1/2, 3/4 give ρ = 7/8, 0, -7/8.
The coefficient values printed above the panels of the two extremizer figures.
The shuffle of M that is countermonotone on [0,3/4] and comonotone on [3/4,1].
Equations
- One or more equations did not get rendered due to their size.
Instances For
The unique maximizer of ρ - η is A_(1/4)^⊥, which is this shuffle of M.