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Optimal multi-weighted step-up procedure for false discovery rate control

Mathematical Methods AIMech:OptimalMultiWeightedStepUpProcedureForFDRControl DRAFT
Aliases: Optimal Multi-Weighted Step-Up Procedure SU(W*), Damped Multi-Weighted Step-Up and Step-Down Procedures SU(W_tilde) and SD(W_tilde), Roquain-van de Wiel Optimal P-Value Weighting, Multi-Weighted Benjamini-Hochberg Procedure
A step-wise multiple hypothesis testing methodology (Roquain & van de Wiel 2009, Electronic Journal of Statistics 3:678-711, arXiv:0807.4081v3, DOI:10.1214/09-EJS430) that generalizes the weighted Benjamini-Hochberg linear step-up procedure LSU(w) by replacing a single static weight vector w = (w_1, ..., w_m) with a rejection-proportion-indexed weight function W(u) = (W_i(u))_{i=1}^m defined across all candidate rejection proportions u in {1/m, 2/m, ..., 1} subject to budget conservation sum_{i=1}^m W_i(u) = m and threshold monotonicity Delta_i(u) = alpha u W_i(u) non-decreasing in u. Because the optimal static weight vector maximizing expected discovery power Pow_w(u) = m^{-1} sum_{i in H_1} F_i(alpha w_i u) depends on the unknown rejection proportion u = |R|/m, Roquain & van de Wiel (2009) solve the constrained Lagrangian maximization simultaneously at every u in (0, 1], yielding the unique optimal weight function W_i^*(u) = (alpha u)^{-1} f_i^{-1}(y^*(u)) under strictly concave alternative CDFs F_i with densities f_i = F_i', where y^*(u) is chosen so that sum_{i=1}^m W_i^*(u) = m (reducing in the one-sided Gaussian model F_i(x) = Phi_bar(Phi_bar^{-1}(x) - mu_i) to alpha u W_i^*(u) = Phi_bar(mu_i / 2 + c(u) / mu_i), Proposition 3.2, Eq. 11). The multi-weighted step-up procedure SU(W) rejects {i : p_i <= alpha u_hat W_i(u_hat)} at u_hat = max{u in {0, 1/m, ..., 1} : G_hat_W(u) >= u}, while the damped procedures SU(W_tilde) with W_tilde_i(u) = W_i(u) / (1 + alpha W_i(1)) and SD(W_tilde) with W_tilde_i(u) = W_i(u) / (1 + alpha u W_i(u)) guarantee exact finite-sample FDR control <= m_0 alpha / m <= alpha under arbitrary deterministic weight functions or misspecified prior parameters (Theorem 4.1), and unmodified SU(W^*) achieves asymptotic FDR = pi_0 alpha and asymptotic power at least as large as every static weighted linear step-up procedure LSU(w) (Theorems 4.2-4.3).
Identity Notes & Concept Scope No normalized identity match for optimal multi-weighted step-up procedure for false discovery rate control or Roquain-van de Wiel optimal p-value weighting across all 442 pinned AIO records and 589 maintained IDs, names, aliases and scoped synonyms. Only frontier reference matches in Round 194 discovery artifacts were found. Distinct from AIMech:AdaptiveDataDrivenOptimalPValueWeighting (Durand 2019, ADDOW / crADDOW), which estimates group null proportions pi_hat_{g,0} and Grenander least-concave-majorant CDFs D_tilde_g directly from the observed p-values over the null-proportion-weighted budget K_hat = {sum_g (m_g/m) pi_hat_{g,0} w_g <= 1}, whereas Roquain & van de Wiel (2009) establish the foundational deterministic individual-hypothesis multi-weighting framework W(u) over sum_{i=1}^m W_i(u) = m, derive the exact closed-form Lagrangian optimal weight function W_i^*(u) = (alpha u)^{-1} f_i^{-1}(y^*(u)) (Proposition 3.2, Eq. 11), prove the m = 2 finite-sample adversarial counterexample for unmodified SU(W) (Lemma C.1), and prove exact non-asymptotic FDR control via damped step-up and step-down weights W_tilde_i(u) (Theorem 4.1) and non-asymptotic Hoeffding oracle inequalities (Theorem 4.2, Proposition 8.3); and distinct from AIMech:CovariatePoweredCrossWeightedMultipleTesting (Ignatiadis & Huber 2021, IHW), which learns static covariate-dependent weights via out-of-fold size-investing.

Claims & Grounded Evidence

4 Claims
Mechanism #lagrangian-optimal-weight-function-and-threshold-monotonicity
Under strictly concave twice-differentiable alternative p-value cumulative distribution functions F_i with decreasing densities f_i = F_i' (Assumptions A1-A3), maximizing the expected discovery power Pow_w(u) = m^{-1} sum_{i in H_1} F_i(alpha w_i u) at each rejection proportion u in (0, 1] subject to sum_{i=1}^m w_i = m yields a unique, continuous optimal weight function W^*(u) = (W_i^*(u))_{i=1}^m given in the unconditional model by W_i^*(u) = (alpha u)^{-1} f_i^{-1}(y^*(u)) (and in the conditional model when alpha < pi_1 by W_i^*(u) = (alpha u)^{-1} f_i^{-1}(y^*(u)) 1{H_i = 1}), where y^*(u) is the unique Lagrange multiplier satisfying sum_{i=1}^m W_i^*(u) = m and the resulting threshold collection Delta_i^*(u) = alpha u W_i^*(u) is strictly increasing in u (Definition 3.1, Proposition 3.2, Section 8.6); in the one-sided Gaussian shift model F_i(x) = Phi_bar(Phi_bar^{-1}(x) - mu_i), the optimal threshold takes the closed form alpha u W_i^*(u) = Phi_bar(mu_i / 2 + c(u) / mu_i) where c(u) in R satisfies m^{-1} sum_{i=1}^m Phi_bar(mu_i / 2 + c(u) / mu_i) = alpha u (Eq. 11), concentrating weight on the largest effect sizes as u -> 0+ and spreading weight across smaller effect sizes as u increases.
Mechanism #finite-sample-step-up-counterexample-and-damped-fdr-control
For m = m_0 = 2 true null hypotheses in the conditional model, the unmodified multi-weighted step-up procedure SU(W) has exact analytical false discovery rate FDR(SU(W)) = alpha + alpha^2 (1 - W_1(1)) (W_1(1) - W_1(1/2)), which attains a maximum of alpha + alpha^2 / 4 > alpha for the valid weight function W_1(1/2) = 0, W_2(1/2) = 2, W_1(1) = 0.5, W_2(1) = 1.5 (Appendix C.1, Lemma C.1), whereas the unmodified step-down procedure SD(W) controls FDR at level alpha whenever m = 2 or m_0 = m (Appendix C.2, Lemma C.2). For arbitrary m >= 1 and any deterministic weight function W, the damped multi-weighted step-up procedure SU(W_tilde) with W_tilde_i(u) = W_i(u) / (1 + alpha W_i(1)) and the damped multi-weighted step-down procedure SD(W_tilde) with W_tilde_i(u) = W_i(u) / (1 + alpha u W_i(u)) guarantee exact non-asymptotic finite-sample FDR <= m_0 alpha / m <= alpha in the conditional model and FDR <= pi_0 alpha <= alpha in the unconditional model under independent p-values, even when the alternative distributions F_i used to construct W are arbitrarily misspecified (Section 4.1, Theorem 4.1, Eq. 13–14; Section 8.1–8.3, Lemmas 8.1–8.2).
Mechanism #finite-sample-oracle-inequality-and-asymptotic-power-dominance
In the unconditional model under strictly concave alternative CDFs F_i, the optimal weight function W^* maximizes the expected rejection curve G_W(u) = pi_0 alpha u + pi_1 Pow_W(u) pointwise at every u in [0, 1], so its fixed point u^* = I(G_{W^*}) satisfies u^* >= u_w = I(G_w) for every static weight vector w; Proposition 8.3 and Theorem 4.2 establish a non-asymptotic finite-sample oracle inequality lower-bounding Pow(SU(W^*)) - max_w Pow(LSU(w)) up to exponentially small Hoeffding concentration terms in m (Eq. 15, Eq. 22–23), and Theorem 4.3 and Proposition 8.4 prove that for any converging weight sequence in W, unmodified SU(W^{*,(m)}) achieves exact asymptotic FDR control lim_{m -> infinity} FDR(SU(W^{*,(m)})) = pi_0 alpha (Eq. 17) and asymptotic power dominance liminf_{m -> infinity} [Pow(SU(W^{*,(m)})) - max_{w^{(m)} in W} Pow(LSU(w^{(m)}))] >= 0 (Eq. 16), automatically matching or exceeding the power of every fixed-u_0 static linear step-up procedure LSU(W^{*,(m)}(u_0)).
Mechanism #prior-effect-size-misspecification-tradeoff-and-step-down-advantage
In finite-sample Gaussian testing with m hypotheses, the damped multi-weighted step-down procedure SD(W_tilde^*) incurs a strictly smaller damping penalty (1 + alpha u W_i^*(u))^{-1} at rejection proportion u < 1 than the damped step-up procedure SU(W_tilde^*)'s uniform penalty (1 + alpha W_i^*(1))^{-1}, yielding higher finite-sample power for SD(W_tilde^*) while outperforming both unweighted LSU and location-only uniform oracle weighting (Section 5.2, Figure 2); under noisy prior effect-size guesses mu_tilde_i = (mu_i + varepsilon_i)_+ with varepsilon_i ~ N(0, sigma^2), both SU(W_tilde(mu_tilde)) and SD(W_tilde(mu_tilde)) maintain exact finite-sample FDR <= alpha for all sigma >= 0 and achieve higher power than LSU for moderate misspecification (sigma in [0, 1.2]), but can lose power relative to unweighted LSU when prior misspecification sigma is large (Section 5.3, Figure 3).

⚠️ Curated Limitations & Negative Results

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes
AIO Target Class Mapping Relation Mapping Rationale
AIO:DataDredgingBias
Data Dredging Bias
 related_match Multi-weighted step-up SU(W*) and step-down SD(W*) procedures control the false discovery rate across heterogeneous hypothesis tests to prevent false-positive inflation (AIO:DataDredgingBias) while optimizing threshold-indexed p-value weights to maximize discovery power.

Evidence Sources & Primary Literature

3 Sources
Source ID Publication / Document Type / Access Reference & Checksum
roquain2009-mwsu-ar5iv
Etienne Roquain, Mark A. van de Wiel
primary paper
full_text
ID: https://doi.org/10.1214/09-EJS430
sha256:118114d17f1c…
roquain2009-mwsu-crossref
Etienne Roquain, Mark A. van de Wiel
primary paper
abstract_only
ID: https://doi.org/10.1214/09-EJS430
sha256:c9bc07d65747…
roquain2009-mwsu-abs
Etienne Roquain, Mark A. van de Wiel
primary paper
abstract_only
ID: https://doi.org/10.48550/arXiv.0807.4081
sha256:cc884794c587…

Ontology & Grounding Topology

Open in interactive network graph →
CONCEPT KIND Mathematical Methods AIMech:OptimalMultiWeightedStepUpProcedureForFDRControl Optimal multi-weighted... 4 claims · 3 sources AIO GROUNDING AIO:DataDredgingBias (1)

Curation Provenance

Curator: Antigravity
Method: ai_assisted
Model: Gemini 3
Created: 2026-10-17T15:05:00Z
Updated: 2026-10-17T15:05:00Z
Source File: data/curated/OptimalMultiWeightedStepUpProcedureForFDRControl.yaml · Schema: curated_concept.yaml

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