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Archimedean copula linear step-up false discovery rate bounds and calibration

Mathematical Methods AIMech:ArchimedeanCopulaStepUpFDRBounds DRAFT

Aliases: Bodnar-Dickhaus Archimedean Copula Linear Step-Up FDR Bound Framework, Archimedean p-Value Copula Benjamini-Hochberg FDR Upper and Lower Bounds, Marshall-Olkin Conditional Order-Statistic Linear Step-Up FDR Calibration

A statistical multiple hypothesis testing and False Discovery Rate (FDR) analysis framework developed by Taras Bodnar and Thorsten Dickhaus (2014, Electronic Journal of Statistics 8(2):2207-2241, arXiv:1305.3897v3, DOI:10.1214/14-EJS950) that models joint p-value distributions P = (P_1, ..., P_m)^T under an Archimedean copula F_P(p_1, ..., p_m) = psi(sum_{i=1}^m psi^{-1}(F_{P_i}(p_i))) with a completely monotone generator psi(t) = E[exp(-t*Z)] via the Marshall-Olkin conditionally independent frailty representation P_i = F_{P_i}^{-1}(psi(-log(Y_i)/Z)): (1) derives a sharper finite-sample upper bound FDR_{vartheta,psi}(phi^{LSU}) <= (m_0(vartheta)/m)*q - b(m, vartheta, psi) for the Benjamini-Hochberg linear step-up test phi^{LSU} (Theorem 3.1, Corollary 3.1) without invoking general MTP_2 inequalities, evaluating the conditional order-statistic probabilities G_k^i(z) under Dirac-uniform configurations via Bolshev's recursion (Equations (26)-(27)); (2) establishes a generator-dependent finite-sample lower bound FDR_{vartheta,psi}(phi^{LSU}) >= (m_0*q/m)*gamma_min(psi) via single-crossing thresholds z^* (Theorem 3.2, Lemma 3.1, Corollary 3.2) with a closed-form incomplete-gamma expression for the Clayton copula (Equations (24)-(25)); (3) characterizes infinite H_0-exchangeable p-value sequences whose m-dimensional marginals are Archimedean copulae possessing the MTP_2 property (Theorem 3.3, Corollary 3.3); and (4) increases the power of phi^{LSU} by pre-estimating the copula parameter eta via pairwise Kendall's tau method-of-moments or maximum likelihood on resampled data and calibrating the nominal level q (Section 5, Equations (34)-(35)).

Identity Notes & Concept Scope Distinct from all 442 pinned AIO concepts and all 597 previously curated AIMech concepts. While AIMech:SelfConsistencyAndDependencyControlFDRFramework (Blanchard & Roquain 2008), AIMech:OneAndTwoStageAdaptiveStepUpFDRUnderDependence (Blanchard & Roquain 2009), AIMech:ContinuumMeasureSpaceStepUpFDRControl (Blanchard, Delattre, & Roquain 2014), and AIMech:StepUpDownLeastFavorableConfigurationFDRBounds (Blanchard, Dickhaus, Roquain, & Villers 2014) analyze FDR control under general PRDS/unspecified dependence or independent two-group mixtures, none of those records exploit the Marshall-Olkin frailty representation Z of Archimedean p-value copulae with completely monotone generators psi(t) = E[exp(-t*Z)] to derive the non-negative conservativity reduction term b(m, vartheta, psi) via Bolshev's order-statistic recursion, the generator-dependent lower bound factor gamma_min(psi) and its Clayton incomplete-gamma closed form, the infinite H_0-exchangeable Archimedean representation (Theorem 3.3), or Kendall's tau copula-calibrated nominal-level adjustment for Clayton and Gumbel-Hougaard families.

Claims & Grounded Evidence

4 Claims

Mechanism #marshall-olkin-sharper-upper-bound-and-direct-fdr-control

Under an Archimedean p-value copula F_P(p_1, ..., p_m) = psi(sum_{i=1}^m psi^{-1}(F_{P_i}(p_i))) with a completely monotone generator psi(t) = E[exp(-t*Z)] (Theorem 2.1, Equation (2)), conditioning on Z = z renders the transformed variables T_j(P_j) = exp(-z*psi^{-1}(F_{P_j}(P_j))) conditionally independent and Uniform[0, 1] under H_j. Theorem 3.1 and Corollary 3.1 (Equations (5)-(9)) prove that for all vartheta in Theta, FDR_{vartheta,psi}(phi^{LSU}) <= (m_0(vartheta)/m)*q - b(m, vartheta, psi) <= (m_0(vartheta)/m)*q, where the non-negative conservativity reduction term is b(m, vartheta, psi) = (q/m) sum_{i in I_0(vartheta)} sum_{k=1}^{m-1} int_{z_k^*}^infty (exp(-z*psi^{-1}(q_{k+1}))/q_{k+1} - exp(-z*psi^{-1}(q_k))/q_k) (G_k^i(z) - G_k^i(z_k^*)) dF_Z(z) >= 0 with single-crossing threshold z_k^* = log(1 + 1/k) / (psi^{-1}(k*q/m) - psi^{-1}((k+1)*q/m)) and conditional order-statistic probability G_k^i(z) = P(bigcap_{r=k}^{m-1} {P_{r:m-1}^{(-i)} > q_{r+1}} | Z = z).

supports Source: False Discovery Rate Control under Archimedean Copula · Locator: Section 2.2, pages 4–5, Theorem 2.1 and Equations (2)–(4); Section 3, pages 5–7, Theorem 3.1, Corollary 3.1, and Equations (5)–(9); Section 7, pages 24–29, Proof of Theorem 3.1
supports Source: False Discovery Rate Control under Archimedean Copula (ar5iv HTML) · Locator: Section 2.2–3, lines 102–207, Theorem 2.1, Theorem 3.1, and Corollary 3.1; Section 7, lines 575–680

Mechanism #generator-dependent-lower-bound-and-clayton-incomplete-gamma-formula

For any Archimedean p-value copula with completely monotone generator psi and continuous marginal p-values on [0, 1], Theorem 3.2 and Lemma 3.1 (Equations (10)-(17)) prove the finite-sample lower bound FDR_{vartheta,psi}(phi^{LSU}) >= (m_0(vartheta)*q/m)*gamma_min(psi), where gamma_min(psi) = 1 - int_0^{z^*} (g(psi^{-1}(q/m)|z) - g(psi^{-1}(q)|z)) dF_Z(z) with g(x|z) = exp(-z*x)/psi(x) and z^* = log(m) / (psi^{-1}(q/m) - psi^{-1}(q)). Corollary 3.2 (Equations (18)-(19)) shows that gamma_min(psi) = 1 when F_Z(z^*) in {0, 1} and gamma_min(psi) -> 1 as m -> infty if (m - 1)*F_Z(z^*(m))/q -> 0. For the Clayton copula psi(t) = (1 + eta*t)^{-1/eta} with Z ~ Gamma(1/eta, eta), Equations (24)-(25) give the exact closed form gamma_min(eta) = 1 + F_{Gamma(1/eta,1)}(ln(m)/(m^eta - 1)) - F_{Gamma(1/eta,1)}(m^eta*ln(m)/(m^eta - 1)).

supports Source: False Discovery Rate Control under Archimedean Copula · Locator: Section 3, pages 7–10, Theorem 3.2, Lemma 3.1, Corollary 3.2, and Equations (10)–(19); Section 4.2, pages 12–14, Equations (22)–(25); Section 7, pages 29–31, Proofs of Theorem 3.2, Lemma 3.1, and Corollary 3.2
supports Source: False Discovery Rate Control under Archimedean Copula (ar5iv HTML) · Locator: Section 3, lines 208–315, Theorem 3.2, Lemma 3.1, and Corollary 3.2; Section 4.2, lines 372–412, Equations (22)–(25); Section 7, lines 681–722

Mechanism #bolshev-recursion-u-shaped-fdr-and-kendall-tau-calibration

Under Dirac-uniform configurations (m_1 = m - m_0 false null p-values equal to 0 almost surely), G_k^i(z) = 1 for k <= m_1 and G_k^i(z) = P_bar_{m_0-1}(a_1, ..., a_{m_0-1}) for k >= m_1 + 1 (where a_j = 0 for j <= k - m_1 - 1 and a_j = exp(-z*psi^{-1}(q_{j+m_1+1})) for j >= k - m_1), which is evaluated in O(m_0^2) operations via Bolshev's recursion P_bar_n(a_1, ..., a_n) = 1 - sum_{j=1}^n comb(n, j) a_j^j P_bar_{n-j}(a_{j+1}, ..., a_n) (Equations (26)-(27)). For the Figure 2 benchmark (m = 20, m_0 = 16, q = 0.05, classical bound 0.04), the sharper Clayton upper bound follows a U-shaped curve across eta (dropping from 0.039856 at eta = 0.05 to 0.030393 at eta = 1.7 and 0.030236 at eta = 2.0 before rising to 0.034237 at eta = 10.0), enabling power gains by pre-estimating eta via pairwise Kendall's tau (tau = eta/(2 + eta) for Clayton; tau = (eta - 1)/eta for Gumbel, Equations (34)-(35)) and increasing the nominal level q so the sharper bound equals target alpha.

supports Source: False Discovery Rate Control under Archimedean Copula · Locator: Section 4.2–4.3, pages 14–17, Equations (26)–(33) and Figures 2–3; Section 5, pages 17–22, Equations (34)–(35) and Figures 4–6
supports Source: False Discovery Rate Control under Archimedean Copula (ar5iv HTML) · Locator: Section 4.2–5, lines 413–554, Equations (26)–(35)

Mechanism #infinite-exchangeable-archimedean-mtp2-characterization

For any infinite sequence (P_i)_{i in N} of absolutely continuous, H_0-exchangeable p-values whose de Finetti-Kingman conditioning variable Z takes values in (0, infty) almost surely with conditional c.d.f. F_{P_tilde_i|Z=z}(p_i) = exp(-z*psi^{-1}(p_i)) where psi(t) = E[exp(-t*Z)] (Equation (21)), Theorem 3.3 and Corollary 3.3 (Equations (20)-(21)) prove that every m-dimensional marginal distribution has the Archimedean copula psi(sum_{i=1}^m psi^{-1}(p_i)) with completely monotone generator psi, possesses the Multivariate Totally Positive of Order 2 (MTP_2) property for all m >= 2, and guarantees exact finite-sample FDR control at level q under phi^{LSU}.

supports Source: False Discovery Rate Control under Archimedean Copula · Locator: Section 3, pages 10–12, Theorem 3.3, Corollary 3.3, and Equations (20)–(21); Section 7, page 31, Proof of Theorem 3.3
supports Source: False Discovery Rate Control under Archimedean Copula (ar5iv HTML) · Locator: Section 3, lines 316–358, Theorem 3.3 and Corollary 3.3; Section 7, lines 723–737

⚠️ Curated Limitations & Negative Results

Practical evaluation of the sharper upper bound via Bolshev's recursion (Equations (26)-(27)) assumes that Dirac-uniform configurations yield upper FDR bounds for phi^{LSU} under Archimedean copula dependence, whereas general finite-sample proofs of the Dirac-uniform least favorable configuration under arbitrary dependence remain open (Section 3, page 7; Section 4.2, page 14).
Pre-estimating the copula parameter eta from N resampled or repeated dataset realizations via Kendall's tau or maximum likelihood requires subset pivotality (the dependence parameter eta of the p-value copula does not depend on the underlying parameter vector vartheta; Section 5, page 17).
For Clayton and Gumbel-Hougaard copulae at fixed positive dependence parameter eta > 0, (m - 1)*F_Z(z^*(m))/q does not vanish as m -> infty and gamma_min(psi) approaches zero at moderate-to-large copula parameters, rendering the lower bound (m_0*q/m)*gamma_min(psi) non-informative away from weak dependence; moreover, along the continuous Clayton Gamma(1/eta, eta) family as eta -> 0+, the threshold z^*(eta) = 1 - eta*ln(sqrt(m)/q) + O(eta^2) approaches 1 from below with (z^*(eta) - 1)/sqrt(eta) -> 0-, so lim_{eta -> 0+} F_Z(z^*(eta)) = 1/2 (whereas at eta = 0, Z == 1 gives F_Z(1) = 1) while lim_{eta -> 0+} gamma_min(eta) = 1 (Section 3, pages 9-10; Section 4.2-4.3, pages 13-16).
All 33 pages of the PDF and all 737 lines of the ar5iv HTML of Bodnar & Dickhaus (2014, Electronic Journal of Statistics 8(2):2207-2241, arXiv:1305.3897v3, DOI:10.1214/14-EJS950) were read. Constructed numerical checks in research/discovery/archimedean-copula-step-up-fdr-bounds-boundaries.py verify (1) Theorem 3.1 and Equations (26)-(27) Bolshev recursion sharper upper bound for the Figure 2 Clayton benchmark (m = 20, m_0 = 16, q = 0.05, classical bound 0.04: ub(0.05) = 0.039856, ub(0.5) = 0.036417, ub(1.7) = 0.030393, minimum ub(2.0) = 0.030236, ub(10.0) = 0.034237), (2) Theorem 3.2 and Equations (24)-(25) Clayton incomplete-gamma lower bound and eta -> 0+ limit (gamma_min(0.05) = 0.738765, lb(0.05) = 0.029551), (3) Gumbel single-crossing threshold z^* (Equation (31): z^*(1.0) = 1.0, z^*(6.6) = 2.23504e-05), and (4) Equations (34)-(35) Kendall's tau calibration identities (Clayton eta = 2: tau = 0.5; Gumbel eta = 2: tau = 0.5).

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	Archimedean copula FDR bounds and copula-calibrated Benjamini-Hochberg step-up testing directly control false discoveries across dependent multiple hypothesis tests, mitigating multiplicity-induced false positives (AIO:DataDredgingBias).

Evidence Sources & Primary Literature

4 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`bodnar-dickhaus2014-pdf`	False Discovery Rate Control under Archimedean Copula Taras Bodnar, Thorsten Dickhaus	primary paper full_text	ID: https://doi.org/10.1214/14-EJS950 sha256:16fbee085fb6…
`bodnar-dickhaus2014-ar5iv`	False Discovery Rate Control under Archimedean Copula (ar5iv HTML) Taras Bodnar, Thorsten Dickhaus	primary paper full_text	ID: https://doi.org/10.1214/14-EJS950 sha256:e227a5c91b32…
`bodnar-dickhaus2014-crossref`	False discovery rate control under Archimedean copula (Electronic Journal of Statistics Record) Taras Bodnar, Thorsten Dickhaus	primary paper abstract_only	ID: https://doi.org/10.1214/14-EJS950 sha256:56e5fff839d0…
`bodnar-dickhaus2014-abs`	False Discovery Rate Control under Archimedean Copula (arXiv:1305.3897 Abstract) Taras Bodnar, Thorsten Dickhaus	primary paper abstract_only	ID: https://doi.org/10.48550/arXiv.1305.3897 sha256:ba7e5b4627fc…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T16:05:00Z

Updated: 2026-10-17T16:05:00Z

Source File: data/curated/ArchimedeanCopulaStepUpFDRBounds.yaml · Schema: curated_concept.yaml

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