The manuscript's parametrization of the extremal families #
In exact-blest-regions.tex every family parameter increases with dependence:
D_c(Section 3, the rank mapζ_ccentred at1-c) runs fromD_0 = WtoD_1 = M;A_w(Section 4.1) pairs the leading fractionwof the first variable comonotonically and runs fromA_0 = WtoA_1 = M;B_b(Section 4.1),b ∈ (0,1/2], splits the leading fractionbbetween two branches, withB_{1/2} = A_{1/2}andB_b → Wasb → 0.
The other modules work with the reflected parameters 1-c, 1-w, and a = 1-b. This module
defines the families exactly as printed (paperD, paperA, paperB) and restates every
family-dependent claim of the manuscript in that notation: the coefficient formulas, parameter
monotonicity and derivatives, graph laws and rank maps, the potentials of Propositions 4.3 and
4.4 with their contact sets, the extremizers in Theorems 1.1 and 1.2 and Corollaries 3.2, 4.5,
and 4.6, Remark 4.7, and all values printed above the panels of Figures 2 and 3.
Reflection of the parameter #
The family A_w #
The manuscript's A_w: under its reflected-coordinate coupling the leading fraction w of
the first variable is comonotone with the trailing fraction of the second, and the remaining
bottom ranks are countermonotone with the top ranks of the second.
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Lemma 4.1: ν(A_w) - η(A_w) = 2(1-w)w³.
Lemma 4.1: w ↦ η(A_w) is strictly increasing.
Lemma 4.1: for w ∈ [1/2,1], η(A_w) ranges over [-3/4,1].
The map T_w of Section 4.1.
A_w is the copula associated with the law of (X, T_w(X)).
The family B_b #
The manuscript's B_b for b ∈ [0,1/2]: the leading fraction b of the first variable is
split between the two branches of R_b, the rest is countermonotone. The value b = 0 gives W,
the continuous endpoint of the family.
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Lemma 4.1: ν(B_b) - η(B_b) = Υ(e_b) with the displayed formula (6).
For b = 1/2, the second branch carries no mass and B_{1/2} = A_{1/2}.
The endpoint b = 0 is W.
Lemma 4.1: b ↦ e_b is strictly increasing on [0,1/2].
Lemma 4.1: e_{1/2} = -3/4 and e_b → -1 as b → 0.
B_b is the copula associated with the law of (R_b(Z), Z).
The two conditional atoms (15) and their probabilities for B_b.
The parameters n_b and Λ #
Section 4.3: b ↦ n_b increases bijectively from (0,1/2) onto (-1,-7/8).
Derivatives of Υ along the parametric branch #
The family D_c #
The manuscript's D_c: the copula of (X, ζ_c(X)) for the rank map ζ_c centred at
1-c, so that D_0 = W, D_1 = M, and the support has its kink at u = c.
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Lemma 3.1, displayed formulas (12).
Lemma 3.1: c ↦ ρ(D_c) is strictly increasing.
Lemma 3.1: c ↦ ρ(D_c) is a bijection of [0,1] onto [-1,1].
Lemma 3.1: ν(D_c) = Φ(ρ(D_c)).
D_c is the copula associated with the law of (X, ζ_c(X)).
Theorem 1.1 and its proof: D_c is the only copula with ρ = ρ(D_c) on the upper
boundary, and its survival copula is the only one on the lower boundary.
Corollary 3.2: the maximum ν - ρ = 1/4 is attained only by D_{1/2}.
Theorem 1.2 in the manuscript's parametrization #
Theorem 1.2 for η ∈ [-3/4,1]: the unique upper extremizer is A_w with
w = ((1+η)/2)^{1/3} ∈ [1/2,1], and the unique lower one is its transpose.
Theorem 1.2 for η ∈ (-1,-3/4): the unique upper extremizer is B_b with e_b = η and
b ∈ (0,1/2), and the unique lower one is its transpose.
Potentials of Propositions 4.3 and 4.4 #
φ_w and ψ_w of (18).
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φ_b and ψ_b of (20).
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Corollaries 4.5 and 4.6 and Remark 4.7 #
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Corollary 4.5: values of A_{3/4}.
Corollary 4.5: |ν - η| = 27/128 exactly for A_{3/4} and A_{3/4}ᵀ.
Corollary 4.6 for n ≥ -7/8: the maximizer is A_wᵀ with w = ((1+n)/2)^{1/4}.
Corollary 4.6 for n < -7/8: the maximizer is B_bᵀ with n_b = n.
Corollary 4.6: the maximizers of νᵀ at given ν > -1 are the transposes A_wᵀ
(w ∈ [1/2,1]) or B_bᵀ (b ∈ (0,1/2)), the minimizers the families themselves.
Remark 4.7(a): for w ∈ (0,1/2), A_w lies strictly below the upper boundary.
Remark 4.7(b): the unique maximizer of ρ - η is A_{3/4}^⊥, the shuffle of M that
is countermonotone on [0,3/4] and comonotone on [3/4,1]; the minimizer is
(A_{3/4}ᵀ)^⊥, its survival copula.
Values printed above the panels of Figures 2 and 3 #
Figure 2 (columns D_{1/4}, D_{1/2}, D_{3/4}) and Figure 3 (columns B_{1/4},
A_{1/2} = B_{1/2}, A_{7/8}), each with the value in the bottom row, and the branch
probabilities 2/3 and 1/3 of B_{1/4}. The decimals printed for B_{1/4} and A_{7/8} are roundings
of these fractions.