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exact step-up and step-down false discovery proportion moment formulas and least favorable configuration analysis

Mathematical Methods AIMech:ExactStepUpStepDownFDPMomentsAndLFCAnalysis DRAFT

Aliases: Roquain-Villers Exact Step-Up and Step-Down FDP Distribution and Moment Framework, Conditional Binomial Step-Up FDP Law and Stirling-Number Moment Decomposition, Step-Down Condition (A) Least Favorable Configuration and Linear Step-Up FDP Variance Extrema

A finite-sample multiple hypothesis testing analysis framework developed by Etienne Roquain and Fanny Villers (2011, Annals of Statistics 39(1):584-612, arXiv:1002.2845v2, DOI:10.1214/10-AOS847) that characterizes the exact joint distribution and moments of the False Discovery Proportion (FDP) for general step-up SU(t) and step-down SD(t) procedures under the unconditional independent two-group mixture model P_bar^I_{(pi_0, F_1)} and equicorrelated multivariate normal (EMN) dependence: (1) proves that conditionally on |SU(t)| = k, the number of false rejections |H_0(H) cap SU(t)| is exactly binomial Binom(k, pi_0*F_0(t_k)/G(t_k)) where G(t) = pi_0*F_0(t) + pi_1*F_1(t) (Theorem 3.1, Equation (11)), yielding exact finite-sample formulas for the FDP cumulative distribution function (Equation (12)), all s-th moments E[FDP(SU(t), H)^s] via Stirling numbers of the second kind {s \atop ell} and step-up rejection-count probabilities D_{m-ell} (Equation (13)), the FDR (Equation (14)), and polynomial-time average power (Equation (15)); (2) proves that |H_0(H) cap SD(t)| given |SD(t)| = k is not binomial for general alternatives F_1 (Equations (16)-(18)) and derives the exact joint rejection distribution (Equation (19)), exact step-down FDR formula (Theorem 3.2, Equation (20)), and closed-form Finner-Roters expansion for linear step-down (LSD) in the Dirac-uniform model (Equation (21)); (3) extends exact SU/SD FDP distribution and moment formulas to non-negatively equicorrelated Gaussian models via 1D/2D Gaussian conditioning (Corollaries 3.4-3.5, Proposition 3.6) and constructs a two-hypothesis negatively correlated Gaussian configuration (m = 2, m_0 = 1, rho -> -1^-, mu = z_1 + z_2) where FDR(LSU) = 3*alpha/4 = alpha*(m_0/m)*(1 + 1/2), attaining the full Benjamini-Yekutieli upper bound and disproving the Dirac-uniform LFC conjecture under negative correlation (Section 4.1.2, Equations (25)-(26)); (4) establishes Condition (A) for step-down procedures to have the Dirac-uniform distribution F_1 == 1 as their least favorable configuration over concave alternative c.d.f.s F_1 in F', proving that LSD satisfies Condition (A) (Theorem 4.1, Equation (24)); and (5) derives exact closed-form minimum and maximum of Var[FDP(LSU)] over general and concave alternative classes F, F', and F_eps (Theorem 4.3, Equations (28)-(30)).

Identity Notes & Concept Scope Distinct from all 442 pinned AIO concepts and all 598 previously curated AIMech concepts. While AIMech:StepUpDownLeastFavorableConfigurationFDRBounds (Blanchard, Dickhaus, Roquain, & Villers 2014) derives upper and lower bounds for general step-up-down procedures of order r via threshold functions tau_k, tau'_k and provenance-independent mixture bounds, and AIMech:ArchimedeanCopulaStepUpFDRBounds (Bodnar & Dickhaus 2014) bounds Benjamini-Hochberg FDR under Archimedean copulae via Marshall-Olkin frailty conditioning, neither record establishes the conditional binomial law |H_0 cap SU(t)| | (|SU(t)| = k) ~ Binom(k, pi_0*F_0(t_k)/G(t_k)) under the unconditional mixture model, the Stirling-number s-th moment formula E[FDP(SU(t))^s] (Theorem 3.1, Equation (13)), the non-binomial conditional law and exact joint distribution for step-down procedures (Theorem 3.2, Equations (16)-(21)), Condition (A) for step-down Dirac-uniform LFCs over concave alternatives and the SU vs SD LFC split on t_k = 0.9*(k/m)^{9/10} (Theorem 4.1), the m = 2 negatively correlated Gaussian counterexample attaining 3*alpha/4 (Equations (25)-(26)), or the closed-form minimum and maximum of Var[FDP(LSU)] over F, F', and F_eps (Theorem 4.3, Equations (28)-(30)).

Claims & Grounded Evidence

4 Claims

Mechanism #conditional-binomial-step-up-fdp-law-and-stirling-moment-formula

Under the unconditional independent two-group mixture model P_bar^I_{(pi_0, F_1)} with marginal c.d.f. G(t) = pi_0*F_0(t) + pi_1*F_1(t), Theorem 3.1 (Equations (11)-(15)) proves that conditionally on |SU(t)| = k (for 1 <= k <= m), the number of false rejections |H_0(H) cap SU(t)| follows the exact binomial distribution Binom(k, pi_0*F_0(t_k)/G(t_k)) (Equation (11)). Consequently, for every integer s >= 1, the s-th moment of the FDP is given exactly by E[FDP(SU(t), H)^s] = sum_{ell=1}^{s wedge m} (m!/(m - ell)!) {s \atop ell} pi_0^ell sum_{k=ell}^m (F_0(t_k)^ell / k^s) D_{m-ell}([G(t_{j+ell})]_{1<=j<=m-ell}, k-ell) (Equation (13)), where {s \atop ell} is the Stirling number of the second kind (correcting Proposition 2.2 of Ferreira & Zwinderman 2006, which omitted {s \atop ell} for s >= 3) and D_m(b, k) = comb(m, k) b_k^k Psi_{m-k}(1 - b_m, ..., 1 - b_{k+1}) is evaluated in O(m^3) operations via Steck's recursion (Equations (6) and (8)).

supports Source: Exact calculations for false discovery proportion with application to least favorable configurations · Locator: Section 2.4, pages 589–590, Equations (6) and (8); Section 3.1.1, pages 591–593, Theorem 3.1 and Equations (11)–(15); Section 6.2, pages 605–606, Proof of Theorem 3.1
supports Source: Exact calculations for false discovery proportion with application to least favorable configurations (ar5iv HTML) · Locator: Section 2.4–3.1.1, lines 80–125, Equations (6), (8), and Theorem 3.1 Equations (11)–(15); Section 6.2, lines 192–198

Mechanism #non-binomial-step-down-conditional-law-and-finner-roters-lsd-fdr

For a step-down procedure SD(t), Section 3.1.2 (Equations (16)-(18)) shows that the conditional distribution of |H_0(H) cap SD(t)| given |SD(t)| = k is not binomial when F_1 is nonlinear (such as F_1(u) = sqrt(u)), and Theorem 3.2 (Equations (19)-(20)) derives the exact step-down FDR formula FDR(SD(t)) = pi_0*m sum_{k=1}^m (t_k/k) D_tilde_{m-1}([G(t_j)]_{1<=j<=m-1}, k-1) sum_{i=0}^{m-k} (k/(k+i)) D_tilde_{m-k}([(G(t_{j+k}) - G(t_k))/(1 - G(t_k))]_{1<=j<=m-k}, i) where D_tilde_m(b, k) = comb(m, k) Psi_k(b_1, ..., b_k) (1 - b_{k+1})^{m-k} (Equation (9)). Under the Dirac-uniform configuration F_1 == 1 and linear thresholds t_k = alpha*k/m, Finner-Roters Lemma 2.1 (Equation (7)) reduces Psi_k(nu_1 + j*nu_2) to (nu_1 + nu_2)*(nu_1 + (k+1)*nu_2)^{k-1}, yielding the closed-form expansion FDR(LSD) = alpha*pi_0 sum_{k=1}^m D_tilde_{m-1}([G(t_j)], k-1) S_{m,k} < alpha*pi_0 (Equation (21)).

supports Source: Exact calculations for false discovery proportion with application to least favorable configurations · Locator: Section 2.4, page 590, Lemma 2.1 and Equations (7) and (9); Section 3.1.2, pages 593–595, Theorem 3.2, Corollary 3.3, and Equations (16)–(21); Section 6.3, page 607, Proof of Theorem 3.2
supports Source: Exact calculations for false discovery proportion with application to least favorable configurations (ar5iv HTML) · Locator: Section 2.4, lines 80–97, Lemma 2.1 and Equations (7), (9); Section 3.1.2, lines 126–146, Theorem 3.2 and Equations (16)–(21); Section 6.3, lines 199–205

Mechanism #condition-a-step-down-concave-lfc-and-negative-correlation-counterexample

Theorem 4.1 (Section 4.1.1, pages 597-599, Equation (24)) proves that if Condition (A) holds—namely k -> (t_k/k) sum_{i=0}^{m-k} (k/(k+i)) D_tilde_{m-k}([(t_{j+k} - t_k)/(1 - t_k)]_{1<=j<=m-k}, i) is nondecreasing for k in {1, ..., m}—then the Dirac-uniform configuration F_1 == 1 maximizes FDR(SD(t)) over all concave alternative c.d.f.s F_1 in F', and proves that the linear step-down procedure t_k = alpha*k/m satisfies Condition (A). For the concave threshold family t_k = 0.9*(k/m)^{9/10} (m = 5, pi_0 = 0.5), k -> t_k/k is decreasing while Condition (A) holds, so the LFC over F_1 in F' is F_1(x) = x for SU(t) (Corollary 4.2) but F_1 == 1 for SD(t). Under Gaussian equicorrelation, Proposition 3.6 and Section 4.1.2 (Equations (25)-(26)) prove continuity in (pi_0, rho, mu) at finite m and show that for m = 2, m_0 = 1, rho -> -1^-, and mu = z_1 + z_2, FDR(LSU) = 3*alpha/4 = alpha*(m_0/m)*(1 + 1/2), attaining the full Benjamini-Yekutieli bound and exceeding the Dirac-uniform level alpha/2.

supports Source: Exact calculations for false discovery proportion with application to least favorable configurations · Locator: Section 3.2, pages 595–597, Corollaries 3.4–3.5 and Proposition 3.6; Section 4.1.1–4.1.2, pages 597–600, Theorem 4.1, Corollary 4.2, and Equations (24)–(26); Section 6.4, pages 607–608
supports Source: Exact calculations for false discovery proportion with application to least favorable configurations (ar5iv HTML) · Locator: Section 3.2–4.1.2, lines 147–178, Corollaries 3.4–3.5, Proposition 3.6, Theorem 4.1, Corollary 4.2, and Equations (24)–(26); Section 6.4, lines 206–232

Mechanism #exact-lsu-fdp-variance-bounds-and-sparse-convergence-rate

For the linear step-up procedure LSU (t_k = alpha*k/m) under the unconditional independent mixture model with F_0(x) = x, Theorem 4.3 and Section 6.5 (Equations (28)-(30) and (35)) prove that E[FDP(LSU, H)^2] = pi_0*alpha*u_0 + (1 - 1/m)*(pi_0*alpha)^2 where u_0(beta, gamma) = sum_{j=0}^{m-1} ((1 - (m - 1 - j)*gamma)/(j + 1)) comb(m - 1, j) (beta - gamma)^j with gamma = pi_0*alpha/m and beta = G(alpha/m), yielding exact closed-form extrema of Var[FDP(LSU, H)]: (i) the minimum over F_1 in F is attained at F_1 == 1 and equals pi_0*alpha*((1 - pi_0*alpha)*((1 - pi1^{m})/(m*pi1)) + (pi_0*alpha/m)*((1 - pi1^{m-1})/pi1 - 1)) (Equation (29)); (ii) the minimum over concave F_1 in F' is attained at F_1(x) = x and equals pi_0*alpha*(1 - alpha)*(1 - (1 - pi_0)*alpha/m); and (iii) the supremum over F_1 in F is approached as F_1 -> 0 and equals pi_0*alpha*(1 - pi_0*alpha) (Equation (30)). In the sparse regime 1 - pi_{0,m} ~ m^{-beta} (0 < beta < 1), min_{F_1 in F} Var[FDP(LSU)] decays only as alpha*m^{-(1-beta)} (at m = 10000, alpha = 0.05, 1 - pi_0 = 0.01, min SD(FDP(LSU)) ~= 0.0217).

supports Source: Exact calculations for false discovery proportion with application to least favorable configurations · Locator: Section 4.2, pages 600–603, Theorem 4.3 and Equations (28)–(30), Figures 3–4; Section 6.5, pages 608–610, Proof of Theorem 4.3 and Equation (35)
supports Source: Exact calculations for false discovery proportion with application to least favorable configurations (ar5iv HTML) · Locator: Section 4.2, lines 179–190, Theorem 4.3 and Equations (28)–(30); Section 6.5, lines 233–291, Equation (35)

⚠️ Curated Limitations & Negative Results

For general non-linear thresholds where Finner-Roters Lemma 2.1 (Equation (7)) does not apply, Steck's recursive computation of Psi_k (Equation (6)) involves alternating-sign binomial terms that suffer severe floating-point cancellation for m >= 50 unless evaluated in high-precision arithmetic (Section 2.4, page 590; Section 3.2, page 596).
Condition (A) in Theorem 4.1 establishes the Dirac-uniform distribution F_1 == 1 as the least favorable configuration for step-down procedures over concave alternative c.d.f.s F_1 in F', whereas for non-concave location-shift alternatives (such as the one-sided Gaussian alternative with moderate mu in Figure 2, page 600) FDR(LSD) can exceed its Dirac-uniform value (Remark 4.1, pages 599-600).
In the proof that the linear step-down procedure satisfies Condition (A) (Section 6.4, page 608), Equation (34) writes S_{m,k} = a_{m,k} + b_{m,k}*S_{m-1,k} by applying ((m-k-i)/(m-k))*comb(m-k, i) = comb(m-k-1, i) while simultaneously replacing the exponent m-k-i of (m - alpha*k - (i+1)*alpha) with m-1-k-i (introducing a small algebraic discrepancy of at most 0.03895035 for m = 6, alpha = 0.2), although direct evaluation of S_{m,k} from Equation (21) confirms that k -> S_{m,k} is strictly increasing ([0.9137931, 0.95108711, 0.97138902, 0.98431953, 0.99333333, 1.0]) so Condition (A) and Theorem 4.1 hold as stated.
All 28 pages of the PDF and all 667 lines of the ar5iv HTML of Roquain & Villers (2011, Annals of Statistics 39(1):584-612, arXiv:1002.2845v2, DOI:10.1214/10-AOS847) were read. Constructed numerical checks in research/discovery/exact-step-up-step-down-fdp-moments-and-lfc-analysis-boundaries.py verify (1) Theorem 3.1 exact agreement of conditional binomial moments (Equation (11)) and Stirling-number moments (Equation (13)) for s = 1..4 at m = 5, pi_0 = 0.6, alpha = 0.25 (E[FDP] = 0.15, E[FDP^2] = 0.1170884792, E[FDP^3] = 0.101398793 vs 0.0875851833 when omitting {3 \atop 2} = 3), (2) Theorem 3.2 non-binomial conditional step-down law (L1 gap 0.12385118) and exact agreement of Equations (20) and (21) for Dirac-uniform LSD (0.1346047661), (3) Proposition 3.6 and Equations (25)-(26) negative-correlation counterexample (FDR(LSU) = 0.0375 = 3*alpha/4 > 0.025), (4) Theorem 4.1 SU vs SD LFC split on t_k = 0.9*(k/m)^{9/10} (SU uniform 0.45956142 > Dirac 0.453016, SD Dirac 0.41564914 > uniform 0.34753598), and (5) Theorem 4.3 exact FDP variance bounds and sparse benchmark at m = 10000, alpha = 0.05, 1 - pi_0 = 0.01 (min SD(FDP) = 0.021685 ~= 0.0217).

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	Exact finite-sample calculations of False Discovery Proportion (FDP) distributions, moments, variances, and least favorable configurations for step-up and step-down procedures directly quantify and control false positive rates in multiple hypothesis testing, mitigating multiplicity-induced false discoveries (AIO:DataDredgingBias).

Evidence Sources & Primary Literature

4 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`roquain-villers2011-pdf`	Exact calculations for false discovery proportion with application to least favorable configurations Etienne Roquain, Fanny Villers	primary paper full_text	ID: https://doi.org/10.1214/10-AOS847 sha256:363776dd96ab…
`roquain-villers2011-ar5iv`	Exact calculations for false discovery proportion with application to least favorable configurations (ar5iv HTML) Etienne Roquain, Fanny Villers	primary paper full_text	ID: https://doi.org/10.1214/10-AOS847 sha256:3e487cf3fae9…
`roquain-villers2011-crossref`	Exact calculations for false discovery proportion with application to least favorable configurations (Annals of Statistics Record) Etienne Roquain, Fanny Villers	primary paper abstract_only	ID: https://doi.org/10.1214/10-AOS847 sha256:80297b7f626c…
`roquain-villers2011-abs`	Exact calculations for false discovery proportion with application to least favorable configurations (arXiv:1002.2845 Abstract) Etienne Roquain, Fanny Villers	primary paper abstract_only	ID: https://doi.org/10.48550/arXiv.1002.2845 sha256:b211939e19d7…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T16:15:00Z

Updated: 2026-10-17T16:15:00Z

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

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