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Papers.AnsariRockel2024.Nelsen7SourceCorrection

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A false intermediate xi integrand in the arXiv v3 appendix #

Appendix A.5.1 of arXiv:2310.17307v3 prints the integrand (θv+1-θ)^2 * (θuv+(1-θ)(u+v-1))₊ for Nelsen 7's xi. The positive part is the CDF, whereas the squared conditional CDF is a step profile. The printed integral fails even for interior parameters. The final Table 6 value xi = 1-θ is proved separately and is correct.

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noncomputable def Papers.AnsariRockel2024.nelsen7PrintedXiCandidate (θ : ↑unitInterval) :
ℝ

The xi expression obtained from the intermediate integrand printed in arXiv:2310.17307v3, Appendix A.5.1. The CDF factor is exactly the printed positive-part expression by Copula.cdf_nelsen7.

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    theorem Papers.AnsariRockel2024.nelsen7_printed_xi_candidate_expanded (θ : ↑unitInterval) :
    nelsen7PrintedXiCandidate θ = (6 * ∫ (v : ↑unitInterval) (u : ↑unitInterval), (↑θ * ↑v + 1 - ↑θ) ^ 2 * max 0 (↑θ * ↑u * ↑v + (1 - ↑θ) * (↑u + ↑v - 1))) - 2

    The candidate is literally the positive-part integrand printed in Appendix A.5.1 of arXiv v3, under the paper's Nelsen 7 CDF convention.

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    theorem Papers.AnsariRockel2024.nelsen7_printed_xi_candidate_formula (θ : ↑unitInterval) :
    nelsen7PrintedXiCandidate θ = -1 - ↑θ / 4

    The printed intermediate expression evaluates to a different affine function on the entire parameter interval.

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    theorem Papers.AnsariRockel2024.nelsen7_printed_xi_candidate_zero :
    nelsen7PrintedXiCandidate 0 = -1

    The printed expression gives -1 at the countermonotonic endpoint.

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    theorem Papers.AnsariRockel2024.nelsen7_printed_xi_identity_false_half :
    (ProbabilityTheory.Copula.nelsen7 Papers.AnsariRockel2024.n7Half✝).chatterjeeXi ≠ nelsen7PrintedXiCandidate Papers.AnsariRockel2024.n7Half✝

    An interior counterexample, within the parameter range of the appendix's calculation: printed xi is -9/8, whereas actual xi is 1/2.

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    theorem Papers.AnsariRockel2024.nelsen7_printed_xi_identity_false (θ : ↑unitInterval) :
    (ProbabilityTheory.Copula.nelsen7 θ).chatterjeeXi ≠ nelsen7PrintedXiCandidate θ

    The false intermediate xi equality fails at every allowed parameter, including the source's entire open interval, while Table 6's final xi formula is correct.