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Papers.AnsariRockel2026RhoFootrule.CorrelationRatio

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Theorem 2.6: outer bounds for Chatterjee xi and the copula correlation ratio #

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noncomputable abbrev Papers.AnsariRockel2026RhoFootrule.conditionalCopies (C : ProbabilityTheory.Copula 2) :
ProbabilityTheory.Copula 2

The conditional-independent-copy copula in equations (38)--(40).

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    noncomputable abbrev Papers.AnsariRockel2026RhoFootrule.copulaCorrelationRatio (C : ProbabilityTheory.Copula 2) :
    ℝ

    Variance of the conditional response-rank mean, normalized by the uniform variance.

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      theorem Papers.AnsariRockel2026RhoFootrule.conditional_copies_cdf (C : ProbabilityTheory.Copula 2) (u v : ↑unitInterval) :
      (conditionalCopies C).cdf ![u, v] = ∫ (t : ↑unitInterval), C.conditionalCDF t u * C.conditionalCDF t v

      Equation (38): the joint distribution of conditional copies.

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      theorem Papers.AnsariRockel2026RhoFootrule.conditional_copies_coefficients (C : ProbabilityTheory.Copula 2) :
      (conditionalCopies C).spearmanFootrule = C.chatterjeeXi ∧ (conditionalCopies C).spearmanRho = copulaCorrelationRatio C

      Equations (39)--(40): both rank coefficients of the actual copy copula.

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      theorem Papers.AnsariRockel2026RhoFootrule.xi_correlationRatio_bounds (C : ProbabilityTheory.Copula 2) :
      max 0 (ProbabilityTheory.Copula.RankRegion.RhoFootrule.lowerBoundary C.chatterjeeXi) ≤ copulaCorrelationRatio C ∧ copulaCorrelationRatio C ≤ min (upperRho C.chatterjeeXi) (2 * C.chatterjeeXi)

      Theorem 2.6, equation (41). No absolute continuity or stochastic monotonicity is assumed.

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      theorem Papers.AnsariRockel2026RhoFootrule.correlationRatio_equality_iff (C : ProbabilityTheory.Copula 2) :
      copulaCorrelationRatio C = 2 * C.chatterjeeXi ↔ C.chatterjeeXi = 0

      Example 2.9: equality in eta <= 2 xi is possible exactly at xi=0.

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      theorem Papers.AnsariRockel2026RhoFootrule.quarter_xi_strict_bound (C : ProbabilityTheory.Copula 2) (h : C.chatterjeeXi = 1 / 4) :
      copulaCorrelationRatio C < 1 / 2

      In particular the outer bound eta=1/2 is unattainable at xi=1/4.