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Definitions for AnsariRockel2024

The measure representation is connected to the classical CDF representation. The paper's FGM formula is identified explicitly. Fréchet weights are on M, W, and independence, in that order; the admissible simplex is a+b ≤ 1. Mardia uses the nonnegative W-weight θ^2*(1-θ)/2; see the source audit.

theorem Papers.AnsariRockel2024.classical_representation (F : (Fin 2 → ↑unitInterval) → ℝ) (hF : ProbabilityTheory.Copula.IsClassical F) : ∃! C : ProbabilityTheory.Copula 2, C.cdf = F

Section 2.1: every classical copula function has exactly one measure representation.

theorem Papers.AnsariRockel2024.fgm_cdf (θ : ℝ) (hθ : |θ| ≤ 1) (u v : ↑unitInterval) : (ProbabilityTheory.Copula.fgm θ hθ).cdf ![u, v] = ↑u * ↑v * (1 + θ * (1 - ↑u) * (1 - ↑v))

Table 5: the FGM CDF on the entire square, for the entire signed parameter range.

theorem Papers.AnsariRockel2024.xi_derivative_formula (C : ProbabilityTheory.Copula 2) : C.chatterjeeXi = (6 * ∫ (v : ↑unitInterval) (u : ↑unitInterval), deriv (C.cdfSection v) ↑u ^ 2) - 2

The source's derivative normalization of xi, without a copula-density assumption.