asymptotically optimal rejection curve false discovery rate control and adjusted step-up-down testing
Mathematical Methods AIMech:AsymptoticallyOptimalRejectionCurveFDRControl DRAFT
Aliases: Finner-Dickhaus-Roters Asymptotically Optimal Rejection Curve (AORC) Framework, Dirac-Uniform Asymptotically Optimal Rejection Curve f_alpha(t) = t / (t*(1 - alpha) + alpha), Adjusted, Truncated, and Step-Up-Down AORC Multiple Testing Procedures
An asymptotic and finite-sample multiple hypothesis testing framework introduced by Helmut Finner, Thorsten Dickhaus, and Markus Roters (2009, Annals of Statistics 37(2):596-618, arXiv:0903.5161v1, DOI:10.1214/07-AOS569) that derives the Asymptotically Optimal Rejection Curve (AORC) f_alpha(t) = t / (t*(1 - alpha) + alpha) for t in [0, 1] with inverse critical value function f_alpha^{-1}(u) = alpha*u / (1 - (1 - alpha)*u) = 1 - f_alpha(1 - u) and induced critical values alpha_{i:n} = i*alpha / (n - i*(1 - alpha)) (Section 3, Equation (3.1), Remark 3.1) by exhausting the False Discovery Rate (FDR) level alpha in (0, 1) across all limiting Dirac-uniform configurations DU_infty(zeta) with limiting empirical c.d.f. F_infty(t | zeta) = (1 - zeta) + zeta*t (zeta in [alpha, 1)). Because alpha_{n:n} = 1 causes an unadjusted step-up procedure based on f_alpha to reject all n hypotheses whenever p_{n:n} < 1, the framework introduces three families of regularized AORC procedures (Examples 3.1-3.3): (1) step-up-down (SUD) procedures phi^lambda of asymptotic order lambda in [0, 1), (2) linearly adjusted step-up procedures based on f^{(1)}_{alpha,kappa} (tangent continuation h_1(x) = f_alpha'(kappa)*(x - kappa) + f_alpha(kappa)) and f^{(2)}_{alpha,kappa} (secant continuation h_2(x) = x*f_alpha(kappa)/kappa), and (3) truncated step-up procedures with critical value curve rho_alpha(x) = min{f_alpha^{-1}(x), kappa}. It establishes general finite-sample FDR upper bounds via conditional rejection-monotonicity conditions (D1)-(D2) and the least isotonic majorant q_bar(x) = max_{0 <= t <= x} rho(t)/t (Section 4, Theorems 4.1-4.3, Equations (4.1)-(4.12)), proves asymptotic FDR control at level alpha with exact limiting FDR formulas at nonzero and zero crossing points r^* (Section 5, Theorems 5.1-5.4, Corollary 5.1, Equations (5.1)-(5.11)), proves that f_alpha is the pointwise minimal rejection curve and maximizes asymptotic power beta_bar_vartheta over the class M_1 of regular asymptotic level-alpha rejection curves (Theorem 5.5, Equations (5.12)-(5.14)), and introduces the finite-sample beta_n-adjusted critical values alpha_{i:n}(beta_n) = i*alpha / (n + beta_n - i*(1 - alpha)) (Section 6, Equation (6.1)).
Identity Notes & Concept Scope
Distinct from all 442 pinned AIO concepts and all 599 previously curated AIMech concepts. While AIMech:ExactStepUpStepDownFDPMomentsAndLFCAnalysis (Roquain & Villers 2011) uses the Finner-Roters order-statistic identity for linear step-down FDR calculations and studies exact finite-sample FDP moments and step-down Condition (A), and AIMech:StepUpDownLeastFavorableConfigurationFDRBounds (Blanchard, Dickhaus, Roquain, & Villers 2014) analyzes finite-sample upper and lower FDR bounds for general step-up-down procedures of order r, neither record defines the Asymptotically Optimal Rejection Curve f_alpha(t) = t / (t*(1 - alpha) + alpha) and its critical values alpha_{i:n} = i*alpha / (n - i*(1 - alpha)) (Equation (3.1)), the tangent- and secant-adjusted rejection curves f^{(1)}_{alpha,kappa} and f^{(2)}_{alpha,kappa} or truncated curve min{f_alpha^{-1}(x), kappa} (Examples 3.1-3.3), the least isotonic majorant finite-sample bound FDR_vartheta <= (n_0/n)*E_{I_0'}[q_bar(R_n/n)] under conditions (D1)-(D2) (Theorems 4.1-4.3), the Glivenko-Cantelli crossing-point asymptotic FDR theorems and pointwise minimality / asymptotic power optimality of f_alpha over M_1 (Theorems 5.1-5.5, Corollary 5.1), or the finite-sample beta_n-adjusted critical value family alpha_{i:n}(beta_n) = i*alpha / (n + beta_n - i*(1 - alpha)) (Equation (6.1)).
Claims & Grounded Evidence
4 Claims
Mechanism
#dirac-uniform-aorc-derivation-and-regularized-rejection-curves
Under the limiting Dirac-uniform configuration DU_infty(zeta) with limiting empirical c.d.f. F_infty(t | zeta) = (1 - zeta) + zeta*t for zeta in [alpha, 1), setting the asymptotic FDR equal to alpha via FDR_zeta(t_zeta) = t_zeta*zeta / ((1 - zeta) + t_zeta*zeta) = alpha yields t_zeta = alpha*(1 - zeta) / (zeta*(1 - alpha)) and uniquely defines the Asymptotically Optimal Rejection Curve (AORC) f_alpha(t) = t / (t*(1 - alpha) + alpha) on [0, 1] with inverse f_alpha^{-1}(u) = alpha*u / (1 - (1 - alpha)*u) = 1 - f_alpha(1 - u) and critical values alpha_{i:n} = f_alpha^{-1}(i/n) = i*alpha / (n - i*(1 - alpha)) (Section 3, pages 599-601, Equation (3.1), Remark 3.1). Because alpha_{n:n} = 1 causes an unadjusted step-up procedure to reject all n hypotheses whenever p_{n:n} < 1, Examples 3.1-3.3 (pages 600-602) construct three regularized AORC families: (1) order-lambda_n(lambda) step-up-down (SUD) procedures phi^lambda with lambda_n(lambda)/n -> lambda in [0, 1), (2) linearly adjusted step-up procedures based on f^{(1)}_{alpha,kappa} (tangent continuation h_1(x) = f_alpha'(kappa)*(x - kappa) + f_alpha(kappa) reaching 1 at x^* = kappa*(1 - alpha)*(2 - kappa) + alpha) and f^{(2)}_{alpha,kappa} (secant continuation h_2(x) = x*f_alpha(kappa)/kappa reaching 1 at x^* = kappa*(1 - alpha) + alpha), and (3) truncated step-up procedures with critical value curve rho_alpha(x) = min{f_alpha^{-1}(x), kappa}.
Mechanism
#finite-sample-fdr-bounds-under-monotonicity-and-isotonic-majorant
Section 4 (pages 602-606) introduces structural test properties (T1)-(T2), conditional rejection-monotonicity conditions (D1)-(D2) (where (D1) requires t -> P_vartheta(R_n >= j | p_i <= t) to be nonincreasing on (0, alpha_{j:n}], which is strictly weaker than PRDS by Wijsman's inequality), and independence conditions (I1)-(I2). Theorem 4.1 (page 603, Equations (4.1)-(4.5)) proves that any multiple test satisfying (T1), (D1), and (D2) with Simes critical values alpha_{i:n} = i*alpha/n satisfies FDR_vartheta(phi_{(n)}) <= (n_0/n)*alpha. For general critical values alpha_{i:n} = rho(i/n) satisfying regularity assumption (A1), Theorem 4.3 (page 605, Equations (4.6)-(4.12)) defines q(x) = rho(x)/x and its least isotonic (nondecreasing) majorant q_bar(x) = max_{0 <= t <= x} q(t) on [0, 1] and proves the universal finite-sample upper bound FDR_vartheta(phi_{(n)}) <= (n_0/n)*E_{vartheta^{i_0}}[q_bar(R_n/n)] <= (n_0/n)*E_{I_0'}[q_bar(R_n/n)] under (T1)-(T2) and (I1)-(I2), with exact equality in the first inequality of Equation (4.7) for step-up tests when (A2) holds (i.e., q(x) = rho(x)/x is nondecreasing on [0, 1]).
Mechanism
#glivenko-cantelli-asymptotic-fdr-control-and-aorc-optimality
Under the asymptotic empirical c.d.f. convergence sup_{t in [0,1]} |F_n(t) - G_vartheta(t)| -> 0 a.s. (Equation (5.2)) with G_vartheta(t) = zeta*t + (1 - zeta)*F_1(t), Theorems 5.1-5.4 (Section 5, pages 606-612, Equations (5.1)-(5.11)) prove that if G_vartheta(rho(x)) has a unique crossing point r^* in (0, f_alpha(t_zeta)] with G_vartheta(rho(x)) > x on [0, r^*) and G_vartheta(rho(x)) < x on (r^*, 1], then R_n/n -> r^* a.s. and lim_{n->infty} FDR_vartheta(phi_{(n)}) = zeta*rho(r^*)/r^* = zeta*q(r^*) <= min{alpha, zeta} (Theorem 5.2), while for r^* = 0 under condition (5.8) or a continuous p.d.f. f_1 on [0, x_0], limsup_{n->infty} FDR_vartheta(phi_{(n)}) <= zeta*q(0) <= zeta*alpha (Theorems 5.3-5.4). Corollary 5.1 (page 611) establishes asymptotic level-alpha FDR control for the SUD procedure phi^lambda (attaining limiting FDR = alpha for all DU_infty(zeta) with zeta in [alpha, 1]) and for the adjusted and truncated step-up procedures based on f^{(1)}_{alpha,kappa}, f^{(2)}_{alpha,kappa}, and min{f_alpha^{-1}(x), kappa} (attaining limiting FDR = alpha for zeta >= zeta^*(kappa) = alpha / (kappa*(1 - alpha) + alpha)). Theorem 5.5 (pages 613-614, Equations (5.12)-(5.14)) proves that every regular rejection curve r in M_1 controlling asymptotic FDR at level alpha satisfies r(t) >= f_alpha(t) on [0, lambda] so inf_{r in M_1} r = f_alpha, and consequently f_alpha maximizes the asymptotic power beta_bar_vartheta(phi) = liminf_{n->infty} E_vartheta[S_n/n_1].
Mechanism
#finite-sample-dirac-uniform-overshoot-and-beta-adjusted-critical-values
Section 6 (pages 614-616, Figure 2 and Equation (6.1)) evaluates the exact finite-sample FDR of the step-up procedure based on the secant-adjusted AORC f^{(2)}_{0.05,kappa_2} (with kappa_2 = (x^* - alpha)/(1 - alpha) = 9/19 ~= 0.47368421 so f^{(2)}_{0.05,kappa_2}(1/2) = 1) under finite-sample Dirac-uniform configurations DU_n(n_0/n) at alpha = 0.05 and n = 100, showing that finite-sample FDR exceeds alpha = 0.05 for intermediate n_0 (attaining its maximum of 0.058013 ~= 0.05801 at n_0 = 16). To enforce strict finite-sample FDR control while preserving convergence to the AORC as n -> infty whenever beta_n = o(n), Equation (6.1) (page 615) introduces the beta_n-adjusted critical values alpha_{i:n}(beta_n) = i*alpha / (n + beta_n - i*(1 - alpha)) for i = 1, ..., n: setting beta_n == 1 recovers the Gavrilov-Benjamini-Sarkar (2009) step-down critical values i*alpha / (n + 1 - i*(1 - alpha)) (which still exceed alpha = 0.05 for a step-up procedure at n = 100 with max FDR 0.060123), whereas setting beta_{100} = 1.76 achieves strict finite-sample step-up FDR control across all n_0 at n = 100, alpha = 0.05 (max FDR 0.049959 <= 0.05).
⚠️ Curated Limitations & Negative Results
- An unadjusted step-up procedure using the raw AORC critical values alpha_{i:n} = i*alpha / (n - i*(1 - alpha)) sets alpha_{n:n} = 1 and therefore rejects all n hypotheses with probability 1 whenever p_{n:n} < 1, yielding FDR = n_0/n > alpha for n_0/n > alpha; asymptotic FDR control requires step-up-down regularization of order lambda < 1, linear tangent/secant adjustment above kappa < 1, or truncation at kappa < 1 (Section 3, Examples 3.1-3.3, pages 600-602).
- Even after linear secant adjustment f^{(2)}_{alpha,kappa_2} with x^* = 1/2, the resulting step-up procedure controls the FDR only asymptotically as n -> infty and exhibits finite-sample FDR overshoot under Dirac-uniform configurations (reaching 0.058013 > 0.05 at n_0 = 16 for n = 100, alpha = 0.05 in Figure 2, page 614) unless damped by a finite-sample shift beta_n = O(n^{1/2+eps}) in Equation (6.1) (Section 6, pages 614-616).
- At the boundary crossing point r^* = 0 (which corresponds to the global null zeta = 1 or vanishing signal), asymptotic FDR control in Theorems 5.3-5.4 requires either the growth condition lim_{n->infty} (1/log n) sum_{i=1}^n (q_bar(i/n) - q(0)) ((i/n)/rho(i/n) - 1) = 0 (Equation (5.8)) or a continuous alternative density f_1 near zero; for exact AORC critical values below kappa, q_bar(x) - q(0) = alpha*(1 - alpha)*x / (1 - (1 - alpha)*x) = O(x) violates Equation (5.8) so a vanishing proportion of Dirac-zero alternatives can still cause local overshoot near zeta = 1 unless regularized (Remark 5.1, pages 611-612; Section 7, pages 616-617).
- All 24 pages of the full-text PDF of Finner, Dickhaus, & Roters (2009, Annals of Statistics 37(2):596-618, arXiv:0903.5161v1, DOI:10.1214/07-AOS569) were read. Constructed numerical checks in research/discovery/asymptotically-optimal-rejection-curve-fdr-control-boundaries.py verify (1) Section 3 AORC identities and symmetry f_alpha^{-1}(t) = 1 - f_alpha(1 - t), (2) Examples 3.2-3.3 and Figure 1 tangent and secant adjustment thresholds (kappa_1 = 0.27452375, kappa_2 = 0.47368421 for x^* = 0.5, alpha = 0.05), (3) Section 6 Figure 2 exact finite-sample Dirac-uniform FDR benchmark at n = 100, alpha = 0.05 via high-precision Bolshev/Steck recursion (reproducing argmax n_0 = 16 and max FDR = 0.058013 matching the paper's reported 0.05801 to five decimal places), (4) Section 6 Equation (6.1) finite-sample beta_n adjustment showing that beta_{100} = 1.76 is the exact two-decimal threshold achieving strict step-up FDR control <= 0.05 at n = 100 (max FDR = 0.049959 vs 0.060123 for beta_n = 1.0), and (5) Corollary 5.1 and Theorem 5.5 asymptotic rejection proportion dominance over Simes LSU (0.78947368 vs 0.75949367 at zeta = 0.25, alpha = 0.05).
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:DataDredgingBias
Data Dredging Bias
|
related_match | Asymptotically optimal rejection curves and regularized step-up-down and step-up critical value families control the False Discovery Rate (FDR) in large-scale simultaneous hypothesis testing while maximizing asymptotic power, directly addressing multiplicity-induced false positives (AIO:DataDredgingBias). |
Evidence Sources & Primary Literature
3 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
finner-dickhaus-roters2009-pdf
|
Helmut Finner, Thorsten Dickhaus, Markus Roters
|
primary paper
full_text
|
ID: https://doi.org/10.1214/07-AOS569
sha256:f93ddc34225f…
|
finner-dickhaus-roters2009-crossref
|
Helmut Finner, Thorsten Dickhaus, Markus Roters
|
primary paper
abstract_only
|
ID: https://doi.org/10.1214/07-AOS569
sha256:f19c0022412b…
|
finner-dickhaus-roters2009-abs
|
On the false discovery rate and an asymptotically optimal rejection curve (arXiv:0903.5161 Abstract)
Helmut Finner, Thorsten Dickhaus, Markus Roters
|
primary paper
abstract_only
|
ID: https://doi.org/10.48550/arXiv.0903.5161
sha256:edcb1378aed6…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini 3
Created: 2026-10-17T16:25:00Z
Updated: 2026-10-17T16:25:00Z
Source File:
data/curated/AsymptoticallyOptimalRejectionCurveFDRControl.yaml
· Schema:
curated_concept.yaml
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