Step-up-down least favorable configuration and two-population Steck false discovery rate bounds
Mathematical Methods AIMech:StepUpDownLeastFavorableConfigurationFDRBounds DRAFT
Aliases: Blanchard-Dickhaus-Roquain-Villers Step-Up-Down LFC Framework, Two-Population Generalized Steck's Recursion for Step-Up-Down FDR, Non-Asymptotic Perturbation Bounds for Step-Up-Down Multiple Testing
A finite-sample and non-asymptotic statistical multiple-testing framework developed by Gilles Blanchard, Thorsten Dickhaus, Etienne Roquain, and Fanny Villers (2014, Statistica Sinica 24(1):1-23, arXiv:1108.5262v1, DOI:10.5705/ss.2011.205) that evaluates and bounds the False Discovery Rate (FDR) of step-up-down procedures SUD_lambda(t) of order lambda in {1, ..., m} under two-group fixed-mixture FM(m, m_0, F) and random-mixture RM(m, pi_0, F) independent p-value models: (1) generalizes Steck's order-statistic recursion to two heterogeneous populations (k_0 uniform true nulls and k - k_0 alternatives with c.d.f. F; Proposition 5.1, Equation (18)) to compute exact finite-sample joint (V, R) distributions and exact FDR(SUD_lambda(t)) (Theorem 5.2, Corollary 5.3); (2) disproves the conjecture (Conjecture 3.2) that the Dirac-uniform configuration (F == 1) is a universal finite-sample least favorable configuration (LFC) for intermediate linear step-up-down procedures LSUD_lambda (Proposition 4.1); and (3) brackets the discrete rejection fraction hat_k/m via the continuous fixed-point functional U(ell, G) (Definition 4.2, Lemma 4.3) to prove explicit Dvoretzky-Kiefer-Wolfowitz perturbation bounds (Theorem 4.4) and O((1 - zeta_m)^{-1} sqrt((log m) / m)) convergence rates (Corollary 4.5) quantifying the maximum excess of FDR(SUD_lambda(t), F) over FDR(SUD_lambda(t), F == 1) for both Simes linear and Finner-Dickhaus-Roters asymptotically optimal rejection curves (AORC).
Identity Notes & Concept Scope
Distinct from all 442 pinned AIO concepts and all 596 previously curated AIMech concepts. While AIMech:SelfConsistencyAndDependencyControlFDRFramework (Blanchard & Roquain 2008), AIMech:OneAndTwoStageAdaptiveStepUpFDRUnderDependence (Blanchard & Roquain 2009), and AIMech:ContinuumMeasureSpaceStepUpFDRControl (Blanchard, Delattre, & Roquain 2014) establish FDR control for step-up and step-down procedures under self-consistency, dependency control, or continuum weak PRDS, none of those records analyze the least favorable alternative configuration (LFC) problem for intermediate step-up-down procedures SUD_lambda(t) (1 < lambda < m), derive the polynomial-time two-population generalized Steck recursion Psi_{k, k_0, F}(t_1, ..., t_k) (Proposition 5.1), disprove the finite-sample Dirac-uniform LFC conjecture for LSUD_lambda (Proposition 4.1), or establish the continuous fixed-point bracketing U(lambda/m, hat_G_m) <= hat_k/m <= U(lambda/m, (hat_G_m + m^{-1}) wedge 1) and DKW perturbation bounds (Lemma 4.3, Theorem 4.4, Corollary 4.5).
Claims & Grounded Evidence
4 Claims
Mechanism
#two-population-steck-recursion-and-exact-sud-fdr
For 0 <= k_0 <= k and independent variables (U_i)_{1 <= i <= k} comprising k_0 i.i.d. Uniform(0, 1) true null p-values and k - k_0 i.i.d. alternative p-values with continuous c.d.f. F, Proposition 5.1 (Equation (18)) proves that the joint order-statistic probability Psi_{k, k_0, F}(t_1, ..., t_k) = P(U_{(1)} <= t_1, ..., U_{(k)} <= t_k) satisfies the polynomial-time recursion Psi_{k, k_0, F}(t_1, ..., t_k) = (t_k)^{k_0} F(t_k)^{k - k_0} - sum_{0 <= j_0 <= j <= k - 2, j_0 <= k_0, j - j_0 <= k - k_0} comb(k_0, j_0) comb(k - k_0, j - j_0) (t_k - t_{j+1})^{k_0 - j_0} (F(t_k) - F(t_{j+1}))^{k - k_0 - j + j_0} Psi_{j, j_0, F}(t_1, ..., t_j) (with Psi_{0,0,F}(.) = 1 and 0^0 = 1), which combines with the step-up/step-down event partition {p_{(lambda)} > t_lambda} cup {p_{(lambda)} <= t_lambda} (Equation (25), Theorem 5.2, and Corollary 5.3, Equations (19)-(27)) to give exact finite-sample formulas for FDR(SUD_lambda(t)) and P(FDP(SUD_lambda(t)) <= x) (Appendix A, Corollary A.1) in both FM(m, m_0, F) and RM(m, pi_0, F).
Mechanism
#intermediate-lsud-dirac-uniform-lfc-counterexample
Although the Dirac-uniform configuration (F == 1, where alternative p-values have point mass 1 at 0) is a finite-sample least favorable configuration for both linear step-up LSU = LSUD_m (Benjamini & Yekutieli 2001) and linear step-down LSD = LSUD_1 over concave F in RM(m, pi_0, F) (Roquain & Villers 2011a), Proposition 4.1 (Equation (8) and Figure 3) proves by exact two-population Steck evaluation at m = 10, level alpha = 0.5, and m_0 = 7 (or pi_0 = 0.7) that for intermediate orders lambda in {4, 5, 6, 7} (and across the complete intermediate set lambda in {3, 4, 5, 6, 7, 8, 9}), the uniform zero-signal alternative F(x) = x yields strictly larger FDR than the Dirac-uniform configuration: FDR(LSUD_lambda, F(x) = x) > FDR(LSUD_lambda, F == 1) in both FM(10, 7, F) (e.g., at lambda = 4: 0.342981 > 0.329850) and RM(10, 0.7, F) (at lambda = 4: 0.342981 > 0.334060), disproving Conjecture 3.2.
Mechanism
#continuous-threshold-bracketing-and-dkw-perturbation-bound
For any continuous non-decreasing critical value function rho: [0, 1] -> [0, 1] such that x -> rho(x)/x is non-decreasing and t_k = rho(k/m), Definition 4.2 and Lemma 4.3 (Equations (9)-(10)) prove that the continuous fixed-point functional U(ell, G) = min{u in [ell, 1] : G(rho(u)) <= u} if G(rho(ell)) >= ell else max{u in [0, ell] : G(rho(u)) >= u} brackets the discrete rejection fraction hat_k / m via U(lambda/m, hat_G_m) <= hat_k / m <= U(lambda/m, (hat_G_m + m^{-1}) wedge 1). Combining this bracketing with Massart's (1990) tight constant 2 in the Dvoretzky-Kiefer-Wolfowitz inequality around G_zeta^{DU}(x) = (1 - zeta) + zeta x, Theorem 4.4 (Equations (11)-(13)) proves that for any F in F, lambda in {1, ..., m}, zeta in (0, 1), and delta in (0, 1), FDR(SUD_lambda(t), m_0, F) <= FDR(SUD_lambda(t), m_0, F == 1) + (m_0/m) varepsilon(delta, m, zeta, nu) in FM(m, m_0, F) (with nu = max_{k in {m_0-1, m_0}} |k/m - zeta|) and FDR(SUD_lambda(t), pi_0, F) <= FDR(SUD_lambda(t), pi_0, F == 1) + pi_0 varepsilon(delta, m, zeta, gamma) + 4 exp(-2 m (gamma - 1/m)_+^2) in RM(m, pi_0, F) (with pi_0 = zeta), where u_delta^+ = U(lambda/m, (G_zeta^{DU} + delta) wedge 1), u_delta^- = U(lambda/m, (G_zeta^{DU} - delta) vee 0), and varepsilon(delta, m, zeta, y) = (rho(u_delta^+) - rho(u_delta^-)) / u_delta^+ + (4 / (1 - zeta)) exp(-2 m (delta - y - 1/m)_+^2 (1 - y/zeta)_+).
Mechanism
#vanishing-rate-for-linear-and-aorc-step-up-down
For Simes linear critical values rho(x) = alpha x (where (rho(u_delta^+) - rho(u_delta^-))/u_delta^+ <= 2 alpha delta / (1 - zeta + delta)) or the Finner-Dickhaus-Roters (2009) asymptotically optimal rejection curve (AORC) rho(u) = alpha u / (1 - u(1 - alpha)) (where (rho(u_delta^+) - rho(u_delta^-))/u_delta^+ ~ 2 alpha (1 - alpha) delta / ((zeta^2 - alpha zeta)(1 - zeta)) as delta -> 0 provided lambda/m < v_delta = 1 - delta alpha / (zeta - alpha) + O(delta^2)), Corollary 4.5 (Equation (15)) proves that when zeta_m in (alpha, 1) satisfies (1 - zeta_m)^{-1} sqrt((log m)/m) = o(1) (and (1 - lambda_m/m)^{-1} sqrt((log m)/m) = o(1) for AORC), the positive part of the FDR excess over the Dirac-uniform model vanishes at the explicit non-asymptotic rate (FDR(SUD_{lambda_m}(t), F) - FDR(SUD_{lambda_m}(t), F == 1))_+ = O((1 - zeta_m)^{-1} sqrt((log m) / m)).
⚠️ Curated Limitations & Negative Results
- As proved in Proposition 4.1 (Section 4.1, Equation (8), Figure 3), the Dirac-uniform configuration (F == 1) is NOT a finite-sample least favorable configuration for intermediate linear step-up-down procedures LSUD_lambda (e.g., for m = 10, m_0 = 7, alpha = 0.5, lambda in {3, 4, 5, 6, 7, 8, 9}), so evaluating finite-sample FDR solely at F == 1 can underestimate the maximum finite-sample FDR over F in F.
- The explicit convergence rate O((1 - zeta_m)^{-1} sqrt((log m) / m)) in Corollary 4.5 requires (1 - zeta_m)^{-1} sqrt((log m) / m) = o(1) and therefore excludes the highly sparse regime where the true-null proportion zeta_m approaches 1 faster than 1 - sqrt((log m) / m) (Section 4.3, page 17).
- For the Finner-Dickhaus-Roters asymptotically optimal rejection curve (AORC) rho(u) = alpha u / (1 - u(1 - alpha)), the perturbation bound in Theorem 4.4 vanishes only when the step-up-down order satisfies lambda_m / m < v_delta = 1 - delta alpha / (zeta - alpha) + O(delta^2), requiring the additional condition (1 - lambda_m / m)^{-1} sqrt((log m) / m) = o(1) (Section 4.2–4.3, pages 16–17).
- All 33 pages of the PDF and all 987 lines of the ar5iv HTML of Blanchard, Dickhaus, Roquain, & Villers (2014, Statistica Sinica 24(1):1-23, arXiv:1108.5262v1, DOI:10.5705/ss.2011.205) were read. Constructed numerical checks in research/discovery/step-up-down-least-favorable-configuration-fdr-bounds-boundaries.py verify (1) Proposition 5.1 two-population generalized Steck recursion and Corollary 5.3 exact finite-sample SUD FDR formulas in FM(10, 7, F) and RM(10, 0.7, F), (2) Proposition 4.1 counterexample across lambda in [3, 4, 5, 6, 7, 8, 9] where FDR(LSUD_lambda, F(x)=x) > FDR(LSUD_lambda, F==1) (at lambda = 4: FM 0.342981 > 0.32985, RM 0.342981 > 0.33406), (3) Lemma 4.3 continuous fixed-point bracketing on the Figure 6 configuration (at lambda = 4: U_lower = 0.4 <= hat_k/m = 0.7 <= U_upper = 0.8), (4) Theorem 4.4 and Corollary 4.5 O(sqrt((log m)/m)) decay across m in {100, 1000, 10000, 100000} (at m = 100000: bound = 0.007018, scaled ratio = 0.654045), and (5) Lemma B.1 extreme configuration where FDR(SD) = 0.790285 > FDR(SU) = 0.7 at t_0 = 0.2 > 0.158018.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:DataDredgingBias
Data Dredging Bias
|
related_match | Step-up-down multiple testing procedures and their least favorable configuration and non-asymptotic perturbation bounds directly control the False Discovery Rate across simultaneous hypothesis tests, preventing multiplicity-induced false positives (AIO:DataDredgingBias). |
Evidence Sources & Primary Literature
4 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
blanchard-dickhaus-roquain-villers2014-pdf
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Gilles Blanchard, Thorsten Dickhaus, Etienne Roquain, Fanny Villers
|
primary paper
full_text
|
ID: https://doi.org/10.5705/ss.2011.205
sha256:3276246bf480…
|
blanchard-dickhaus-roquain-villers2014-ar5iv
|
Gilles Blanchard, Thorsten Dickhaus, Etienne Roquain, Fanny Villers
|
primary paper
full_text
|
ID: https://doi.org/10.5705/ss.2011.205
sha256:ff44c60469e2…
|
blanchard-dickhaus-roquain-villers2014-crossref
|
Gilles Blanchard, Thorsten Dickhaus, Etienne Roquain, Fanny Villers
|
primary paper
abstract_only
|
ID: https://doi.org/10.5705/ss.2011.205
sha256:b5a8c00ddcee…
|
blanchard-dickhaus-roquain-villers2014-abs
|
Gilles Blanchard, Thorsten Dickhaus, Etienne Roquain, Fanny Villers
|
primary paper
abstract_only
|
ID: https://doi.org/10.48550/arXiv.1108.5262
sha256:3a141ad1099f…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini 3
Created: 2026-10-17T15:55:00Z
Updated: 2026-10-17T15:55:00Z
Source File:
data/curated/StepUpDownLeastFavorableConfigurationFDRBounds.yaml
· Schema:
curated_concept.yaml
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