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Hadamard-differentiable threshold functional false discovery proportion central limit theorems and adaptive fixed-point iteration duality

Mathematical Methods AIMech:HadamardDifferentiableThresholdFDPAsymptoticsAndFixedPointDuality DRAFT

Aliases: Neuvial Functional Delta-Method FDP Asymptotics and Threshold Function Framework, Hadamard-Differentiable Threshold Function T(F) = U(F, A(F)) Central Limit Theorem, Fixed-Point Iteration Duality Between One-Stage (FDR08, BR08) and Two-Stage (Sto02, BKY06) Adaptive FDR Procedures

An asymptotic empirical-process and functional Delta-method framework introduced by Pierre Neuvial (2008, Electronic Journal of Statistics 2:1065-1110, arXiv:0803.2111v2, DOI:10.1214/08-EJS207) that characterizes the limiting distribution of the random rejection threshold T(G_hat_m) and the achieved False Discovery Proportion FDP_m(T(G_hat_m)) = V(G_hat_{0,m}, G_hat_{1,m}) / (R(pi_0*G_hat_{0,m} + (1 - pi_0)*G_hat_{1,m}) vee (1/m)) of multiple testing procedures under independent p-values. Representing any threshold procedure on cadlag distribution functions D[0, 1] as T(F) = U(F, A(F)) with U(F, alpha) = sup{u in [0, 1] : F(u) >= r_alpha(u)}, rejection curve r_alpha, and level function A : D[0, 1] -> [0, 1] (Definitions 2.1-2.2, Section 2.2), Theorem 3.2 and Theorem 7.13 prove that when T has a unique interior right crossing point tau^* = T(G) in (0, 1) where G(tau^*) = r(A(G), tau^*) and g(tau^*) < d_u r(A(G), tau^*) (Definition 3.7, Conditions C.1-C.3), T is Hadamard-differentiable at G tangentially to C[0, 1] with derivative T_dot_G(H) = (H(tau^*) - d_alpha r(A(G), tau^*)*A_dot_G(H)) / (d_u r(A(G), tau^*) - g(tau^*)) (Proposition 7.11, Corollary 7.12), and sqrt(m)*(FDP_m(T(G_hat_m)) - p(tau^*)) converges in distribution to the centered Gaussian limit X = p^*(1 - p^*)*(Z_0(tau^*)/tau^* - Z_1(tau^*)/G_1(tau^*)) + p_dot(tau^*)*T_dot_G(Z), while o(m^{-1/2}) threshold perturbations are asymptotically equivalent (Definition 3.5, Proposition 3.6). Applying this theorem yields explicit Central Limit Theorems and asymptotic power comparisons for BH95, BH95 Oracle, one-stage adaptive procedures FDR08(lambda) and BR08(lambda), and two-stage plug-in procedures Sto02(lambda)/STS04(lambda) and BKY06(lambda) (Section 4, Theorems 4.2, 4.7, 4.10, 4.12, 4.15, Corollary 4.16, Theorem 4.20, Table 1), and establishes a formal fixed-point iteration duality proving that the asymptotic threshold of the one-stage procedure FDR08 is the fixed point of iterating the two-stage procedure Sto02(lambda) and the asymptotic threshold of BR08(lambda) is the fixed point of iterating BKY06(lambda) (Section 5, Theorems 5.2, 5.4, 5.6, Corollaries 5.3, 5.5, 5.7).

Identity Notes & Concept Scope Distinct from all 442 pinned AIO concepts and all 601 previously curated AIMech concepts. While AIMech:OneAndTwoStageAdaptiveStepUpFDRUnderDependence (Blanchard & Roquain 2008) introduces the finite-sample one-stage procedure BR08(lambda) and studies finite-sample FDR control of one- and two-stage adaptive procedures under independence and dependence, AIMech:AsymptoticallyOptimalRejectionCurveFDRControl (Finner, Dickhaus, & Roters 2009) derives the AORC f_alpha(u) = u / (alpha + (1 - alpha)*u) (procedure FDR08) and its almost-sure crossing-point FDR limit, and AIMech:ExactStepUpStepDownFDPMomentsAndLFCAnalysis (Roquain & Villers 2011) analyzes exact finite-sample moments of the FDP, none of those records formulate the Hadamard-differentiable threshold functional T(F) = U(F, A(F)) on D[0, 1] and the functional Delta-method Central Limit Theorem for sqrt(m)*(FDP_m(T(G_hat_m)) - p(tau^*)) (Theorems 3.2, 7.3, 7.13, Proposition 7.11, Corollary 7.12), the o(m^{-1/2}) perturbation asymptotic equivalence criterion (Proposition 3.6, Proposition 7.16), the explicit Gaussian limit distributions and variances of BH95, BH95o, FDR08(lambda), BR08(lambda), Sto02(lambda)/STS04(lambda), and BKY06(lambda) (Theorems 4.2, 4.7, 4.10, 4.12, 4.15, Corollary 4.16, Theorem 4.20), or the fixed-point iteration duality Theorems 5.2, 5.4, and 5.6 linking Sto02(lambda) to FDR08 and BKY06(lambda) to BR08(lambda) with Corollaries 5.3, 5.5, and 5.7.

Claims & Grounded Evidence

4 Claims

Mechanism #hadamard-differentiable-threshold-functional-and-fdp-central-limit-theorem

Writing the False Discovery Proportion achieved by a threshold functional T : D[0, 1] -> [0, 1] as FDP_m(T(G_hat_m)) = V(G_hat_{0,m}, G_hat_{1,m}) / (R(pi_0*G_hat_{0,m} + (1 - pi_0)*G_hat_{1,m}) vee (1/m)) with V(F_0, F_1) = pi_0*F_0(T(pi_0*F_0 + (1 - pi_0)*F_1)) and R(F) = F(T(F)) (Section 2.2, pages 1068-1071), Theorem 3.2 (pages 1072-1073) and Theorem 7.3 (page 1092) prove via Donsker's invariance principle (Theorem 3.1) and the functional Delta method that if tau^* = T(G) > 0 and T is Hadamard-differentiable at G tangentially to C[0, 1] with derivative T_dot_G (Condition C.1), then sqrt(m)*(T(G_hat_m) - tau^*) ->_d T_dot_G(Z), lim_{m->infty} FDR_m(T(G_hat_m)) = p(tau^*) = pi_0*tau^*/G(tau^*), and sqrt(m)*(FDP_m(T(G_hat_m)) - p(tau^*)) ->_d X = p(tau^*)*(1 - p(tau^*))*(Z_0(tau^*)/tau^* - Z_1(tau^*)/G_1(tau^*)) + p_dot(tau^*)*T_dot_G(Z). For threshold functionals T(F) = U(F, A(F)) with U(F, alpha) = sup{u in [0, 1] : F(u) >= r_alpha(u)}, Proposition 7.11, Corollary 7.12, and Theorem 7.13 (pages 1099-1101) prove that if T has a unique interior right crossing point tau^* in (0, 1) with G(tau^*) = r(A(G), tau^*) and g(tau^*) < d_u r(A(G), tau^*) (Conditions C.2-C.3), r is C^1, and A is Hadamard-differentiable at G, then T_dot_G(H) = (H(tau^*) - d_alpha r(A(G), tau^*)*A_dot_G(H)) / (d_u r(A(G), tau^*) - g(tau^*)), and any perturbed procedure T_m satisfying T(F - eps_m) <= T_m(F) <= T(F) with eps_m = o(m^{-1/2}) has identical asymptotic FDP distribution (Proposition 3.6, pages 1073-1074).

supports Source: Asymptotic properties of false discovery rate controlling procedures under independence · Locator: Section 2.2, pages 1068–1071; Section 3, pages 1071–1075, Condition C.1, Theorems 3.1–3.2, Corollary 3.3, Definitions 3.5 and 3.7, Proposition 3.6, and Conditions C.2–C.3; Section 7.1–7.2, pages 1090–1101, Propositions 7.2, 7.5, 7.11, Corollary 7.12, and Theorems 7.3 and 7.13
supports Source: Asymptotic properties of false discovery rate controlling procedures under independence (arXiv:0803.2111 Abstract) · Locator: Abstract (stochastic process and functional Delta-method derivation of generic Central Limit Theorems for the FDP and associated regularity conditions)

Mechanism #bh95-criticality-and-one-stage-adaptive-fdr08-br08-asymptotic-distributions

For the BH95 procedure (r_alpha(u) = u/alpha, A = alpha), Condition C.2 holds iff alpha > alpha^* = inf_{u->0} u/G(u) = lim_{u->0} 1/g(u) (Condition C.4, page 1075, illustrated in Figure 2, page 1076 for Laplace location statistics with theta = 2, pi_0 = 0.5 where alpha^* = 2/(1 + e^2) ~= 0.23840584), and because d_u r_alpha(tau^*) = 1/alpha = G(tau^*)/tau^* implies zeta(tau^*) = 0, the alternative empirical-process fluctuation Z_1(tau^*) cancels out of the FDP limit: sqrt(m)*(FDP_m(T^{BH95}(G_hat_m)) - pi_0*alpha) ->_d N(0, (pi_0*alpha)^2*(1 - tau^*)/tau^*) (Theorem 4.2, pages 1075-1076; Oracle BH95o in Corollary 4.3, pages 1076-1077). For the one-stage AORC procedure FDR08(lambda) truncated at lambda >= kappa = alpha*(1 - pi_0)/((1 - alpha)*pi_0) (Definition 4.5, Condition C.6), all lambda in [kappa, 1) are asymptotically equivalent (Proposition 4.6), and under Conditions C.3, C.4, and C.7 (alpha < pi_0), Theorem 4.7 (page 1078) proves asymptotic mean p^* = alpha*pi_0 / pi_0_bar(tau^*) in [pi_0*alpha, alpha] (where pi_0_bar(t) = (1 - G(t))/(1 - t)) and limit X^{FDR08} with zeta(tau^*) = -(1 - pi_0_bar(tau^*))*pi_0_bar(tau^*)/alpha / (pi_0_bar(tau^*)^2/alpha - g(tau^*)). For the one-stage procedure BR08(lambda) with r_alpha(u) = b_alpha^lambda(u) = u / (alpha*(1 - lambda) + u) on [0, lambda] (Definition 4.8), Theorem 4.10 (page 1079) proves asymptotic mean p^* = alpha*pi_0*(1 - lambda)/(1 - G(tau^*)) <= alpha under Conditions C.3, C.8 (alpha*(1 - lambda) > alpha^*), and C.9 (G(lambda) <= (lambda/alpha)*(1 - G(lambda))/(1 - lambda)), and shows that BR08(lambda) >> BH95 iff tau^*_{BH95} >= alpha*lambda (page 1080).

supports Source: Asymptotic properties of false discovery rate controlling procedures under independence · Locator: Section 4.1–4.2, pages 1075–1080, Figure 2, Conditions C.4–C.9, Theorem 4.2, Corollary 4.3, Proposition 4.6, Theorems 4.7 and 4.10, and Section 7.3.1, pages 1102–1103, Theorem 7.14

Mechanism #two-stage-plug-in-sto02-sts04-and-bky06-asymptotic-distributions

For two-stage Simes-line plug-in procedures T(F) = sup{u in [0, 1] : F(u) >= u / A(F)} (Section 4.3, Definition 4.11, page 1080), Theorem 4.12 proves that if A is Hadamard-differentiable at G and A(G) > alpha^* (Condition C.10), then sqrt(m)*(FDP_m(T(G_hat_m)) - pi_0*A(G)) ->_d pi_0*A(G)*(Z_0(tau^*)/tau^* + A_dot_G(Z)/A(G)). For Storey's procedure Sto02(lambda) with A(F) = alpha / pi_0_bar^F(lambda) = alpha*(1 - lambda)/(1 - F(lambda)) and the finite-sample conservative variant STS04(lambda), Proposition 7.16 (page 1104) proves asymptotic equivalence under Conditions C.9 and C.11 (alpha > pi_0_bar(lambda)*alpha^*), and Theorem 4.15 and Corollary 4.16 (pages 1081-1082) prove that sqrt(m)*(FDP_m(T^{Sto02}(G_hat_m)) - (pi_0/pi_0_bar(lambda))*alpha) ->_d X^{Sto02} ~ N(0, Var(X^{Sto02})) where for tau^* <= lambda, Var(X^{Sto02}(lambda)) = (pi_0*alpha/pi_0_bar(lambda))^2 * {(1 - tau^*)/tau^* + Var(Z(lambda))/(1 - G(lambda))^2 + 2*pi_0/pi_0_bar(lambda)}. For the two-stage procedure BKY06(lambda) with A(F) = alpha*(1 - lambda)/(1 - F(U(F, lambda))) and first-stage BH95 threshold u(lambda) = U(G, lambda) (Definition 4.17), Theorem 4.20 (page 1083) proves under Conditions C.8 and C.12 (lambda > alpha^*) that the asymptotic mean FDP is pi_0*alpha*(1 - lambda)/(1 - G(u(lambda))) <= alpha with limit X^{BKY06(lambda)} = (pi_0*alpha*(1 - lambda)/(1 - G(u(lambda)))) * (Z_0(tau^*)/tau^* + Z(u(lambda)) / ((1 - alpha*(1 - lambda)*g(u(lambda)))*(1 - G(u(lambda))))), and shows that BKY06(lambda) >> BH95 iff G(u_lambda) >= lambda iff G^2(u_lambda) >= u_lambda.

supports Source: Asymptotic properties of false discovery rate controlling procedures under independence · Locator: Section 4.3, pages 1080–1083, Definitions 4.11, 4.13–4.14, 4.17, Conditions C.10–C.12, Theorems 4.12, 4.15, Corollary 4.16, Theorem 4.20, and Section 7.3.2, pages 1104–1106, Propositions 7.15–7.17

Mechanism #fixed-point-iteration-duality-between-one-stage-and-two-stage-adaptive-procedures

Section 5 (pages 1083-1087) establishes a formal duality between one-stage adaptive procedures T(F) = sup{u in [0, lambda] : F(u) >= c_alpha(F, u)} satisfying Condition C.13 (u -> c_alpha(G, u)/u is nonincreasing on [0, lambda]) and the associated family of two-stage Simes-line procedures T_t(F) = sup{u in [0, 1] : F(u) >= (c_alpha(F, t)/t)*u} with asymptotic thresholds tau(t) = T_t(G). Theorem 5.2 (pages 1085-1086) proves that (i) t <= tau^* implies tau(t) in [t, tau^*] and t >= tau^* implies tau(t) in [tau^*, t], and (ii) for any initial parameter t_0 = t in (0, lambda], the iterated sequence t_{i+1} = tau(t_i) converges monotonically to the one-stage asymptotic threshold lim_{n->infty} t_n = tau^* = T(G), while Corollary 5.3 proves that T_t >> T iff tau(t) > tau^* iff t > tau^* iff t > tau(t). Applying Theorem 5.2 with c_alpha(F, u) = (u/alpha)*(1 - F(u))/(1 - u) proves that the asymptotic threshold of procedure FDR08 is the exact fixed point of iterating Sto02(lambda) (Theorem 5.4, page 1086) and Sto02(lambda) >> FDR08 iff lambda > tau^{Sto02}(lambda) (Corollary 5.5, page 1087). Symmetrically, applying Theorem 5.2 with c_alpha(F, u) = (u/alpha)*(1 - F(u))/(1 - lambda) proves that the asymptotic threshold of procedure BR08(lambda) is the exact fixed point of iterating BKY06(lambda) starting from t_0 = u(lambda) (Theorem 5.6, page 1087) and BR08(lambda) >> BKY06(lambda) iff tau^*_{BR08(lambda)} >= lambda - alpha*(1 - lambda) (Corollary 5.7, which always holds when lambda <= alpha/(1 + alpha)).

supports Source: Asymptotic properties of false discovery rate controlling procedures under independence · Locator: Section 5, pages 1083–1087, Condition C.13, Theorems 5.2, 5.4, 5.6, Corollaries 5.3, 5.5, 5.7, Section 6, pages 1088–1089, Table 1, and Section 7.4, pages 1106–1107
supports Source: Asymptotic properties of false discovery rate controlling procedures under independence (arXiv:0803.2111 Abstract) · Locator: Abstract (interpretation of one-stage adaptive procedures BR08 and FDR08 as fixed points in the iteration of two-stage adaptive procedures BKY06 and Sto02)

⚠️ Curated Limitations & Negative Results

The functional Delta-method Central Limit Theorem (Theorem 3.2) requires the existence and uniqueness of a strictly positive interior right crossing point tau^* = T(G) > 0 (Conditions C.1-C.3); when the target FDR level alpha falls below the critical value alpha^* = 1/g(0) (as in the Laplace location model of Figure 2 on page 1076 where alpha^* = 0.23840584 for theta = 2, pi_0 = 0.5), tau^* = 0, the number of rejections R_m is stochastically bounded as m -> infty, and the sqrt(m) Gaussian limit does not apply (Sections 3.3, 4.1, and 6, pages 1074-1076, 1088-1089).
Unlike FDR08 and Sto02(lambda)/STS04(lambda), the adaptive procedures BR08(lambda) and BKY06(lambda) are not universally more powerful than BH95 asymptotically: BR08(lambda) >> BH95 iff tau^*_{BH95} >= alpha*lambda (page 1080), and BKY06(lambda) >> BH95 iff G(u_lambda) >= lambda (so whenever G(x) <= sqrt(x) on [0, 1], BH95 is strictly more powerful than BKY06(lambda) for every lambda in (0, 1), pages 1083 and 1089).
Three minor typographical slips in the primary source were identified and verified during full-text inspection: (1) in the last displayed equation of the proof of Proposition 7.5 (page 1097), the '+' sign between nabla_F U_{(G, A(G))}(H) and A_dot_G(H)*nabla_alpha U_{(G, A(G))} is omitted (whereas the statement of Proposition 7.5 at the top of page 1097 writes the sum correctly), (2) in the statement of Theorem 7.13 (page 1101, line 4), the second term of X is printed with a '+' sign before p^*(1 - p^*)*zeta(tau^*)*Z_1(tau^*)/G_1(tau^*) whereas the proof of Theorem 7.13 at the bottom of page 1101 and Theorems 4.7, 4.10, and 7.14 all have the correct '-' sign, and (3) in Table 2 (page 1108), the row for G_hat_m writes pi_0*G_hat_{0,m} + G_hat_{1,m}, omitting the factor (1 - pi_0) written on pages 1069-1070.
All 46 pages of the full-text PDF of Neuvial (2008, Electronic Journal of Statistics 2:1065-1110, arXiv:0803.2111v2, DOI:10.1214/08-EJS207) were read. Constructed numerical checks in research/discovery/hadamard-differentiable-threshold-fdp-asymptotics-and-fixed-point-duality-boundaries.py verify (1) Section 4.1 Figure 2 Laplace criticality value alpha^* = 1/g(0) = 0.23840584 at theta = 2, pi_0 = 0.5, (2) Section 4 and Table 1 exact asymptotic thresholds, attained asymptotic FDRs, and asymptotic FDP variances under the Gaussian location model (pi_0 = 0.6, mu = 2.5, alpha = 0.1, lambda = 0.5) together with the Section 4.3 power reversal G(x) <= sqrt(x) where BH95 dominates BKY06(lambda) (tau^*_{BH95} = 0.00277008 > tau^*_{BKY06} = 0.00110293), and (3) Section 5 Theorems 5.2, 5.4, and 5.6 exact fixed-point convergence of iterated Sto02(t_n) to tau^*_{FDR08} = 0.052192013 and iterated BKY06(lambda) to tau^*_{BR08(lambda)} = 0.0488615394 at lambda = alpha/(1 + alpha).

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	Hadamard-differentiable threshold functional Central Limit Theorems and adaptive fixed-point iteration duality characterize the asymptotic fluctuations of the False Discovery Proportion and asymptotic power of multiple testing procedures under independence, directly addressing multiplicity-induced false positives (AIO:DataDredgingBias).

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`neuvial2008-pdf`	Asymptotic properties of false discovery rate controlling procedures under independence Pierre Neuvial	primary paper full_text	ID: https://doi.org/10.1214/08-EJS207 sha256:2cceab6ae1b1…
`neuvial2008-crossref`	Asymptotic properties of false discovery rate controlling procedures under independence (Electronic Journal of Statistics Record) Pierre Neuvial	primary paper abstract_only	ID: https://doi.org/10.1214/08-EJS207 sha256:7a40c41d638f…
`neuvial2008-abs`	Asymptotic properties of false discovery rate controlling procedures under independence (arXiv:0803.2111 Abstract) Pierre Neuvial	primary paper abstract_only	ID: https://doi.org/10.48550/arXiv.0803.2111 sha256:d2d2a7c0253f…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T16:45:00Z

Updated: 2026-10-17T16:45:00Z

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

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