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Measure-theoretic step-up false discovery rate control over a continuum of null hypotheses

Mathematical Methods AIMech:ContinuumMeasureSpaceStepUpFDRControl DRAFT

Aliases: Blanchard-Delattre-Roquain Continuum Step-Up FDR Control, Weakly PRDS Continuum Multiple Testing Framework, Continuous-Index Measure-Space Step-Up Procedure

A measure-theoretic multiple hypothesis testing framework developed by Gilles Blanchard, Sylvain Delattre, and Etienne Roquain (2014, Bernoulli 20(1):304-333, arXiv:1110.3599v3, DOI:10.3150/12-BEJ488) that extends step-up False Discovery Rate control to an arbitrary (possibly uncountably infinite) measurable index space of hypotheses (H, frak_H, Lambda) with finite positive measure Lambda(H) < infty: given a measurable p-value process (p_h(X))_{h in H} and threshold collection Delta(h, r) = alpha pi(h) beta(r), the continuum step-up procedure rejects R_{SU}(X) = L_Delta(r_hat(X)) where L_Delta(r) = {h in H : p_h(X) <= Delta(h, r)} and r_hat(X) = max{r in [0, Lambda(H)] : Lambda(L_Delta(r)) >= r} (satisfying the exact fixed-point volume identity Lambda(L_Delta(r_hat(X))) = r_hat(X) when Delta is left-continuous in r), and controls FDR(R_{SU}) = E[(Lambda(R_{SU} cap H_0) / Lambda(R_{SU})) 1{Lambda(R_{SU}) > 0}] <= alpha int_{H_0} pi d_Lambda either (1) under linear shape beta(r) = r when the p-value process is weakly Positive Regression Dependent on a Subset (weakly PRDS on H_0, meaning all finite marginals (p_h)_{h in F} are PRDS on F cap H_0; Definition 2.4 and Theorem 4.1) via Glivenko-Cantelli empirical measure approximations Lambda_n -> Lambda (Section 5.1, Lemmas 5.1-5.3), or (2) under arbitrary dependence with prior-integrated shape beta_nu(r) = int_0^r u d_nu(u).

Identity Notes & Concept Scope Distinct from all 442 pinned AIO concepts and all 595 previously curated AIMech concepts. While AIMech:SelfConsistencyAndDependencyControlFDRFramework (Blanchard & Roquain 2008, Electronic Journal of Statistics 2:963-992) formulates Self-Consistency and Dependency Control on countable hypothesis spaces H (and only remarks that definitions can be stated on continuous spaces), Blanchard, Delattre, & Roquain (2014, Bernoulli 20(1):304-333, arXiv:1110.3599v3, DOI:10.3150/12-BEJ488) establish the rigorous measure-theoretic foundations for uncountable hypothesis spaces (H, frak_H, Lambda): joint measurability of the continuum step-up volume r_hat(X) = max{r in [0, Lambda(H)] : Lambda(L_Delta(r)) >= r} (Lemma 3.1), equivalence of four formulations of weak PRDS on infinite index sets via Strassen's theorem and the Portmanteau theorem (Definitions 2.3-2.4, Proposition 2.2), proof of continuum step-up FDR control via Glivenko-Cantelli empirical measure discretization Lambda_n -> Lambda (Theorem 4.1, Lemmas 5.1-5.3), and exact finite-dimensional computational reductions for continuous c.d.f. testing gamma(u) = sum_{j=0}^m (t_j(u) - X_{(j)})_+ wedge (X_{(j+1)} - X_{(j)}) (Corollary 4.4, Remark 4.6) and Poisson process intensity testing with data-dependent interval-length weights (Corollary 4.5, Remark 4.6).

Claims & Grounded Evidence

4 Claims

Mechanism #equivalence-of-infinite-dimensional-weak-prds-characterizations

For an infinite family of p-value processes p = (p_h(X))_{h in H} indexed by a measurable space (H, frak_H), Proposition 2.2 proves via Strassen's monotone coupling theorem and the Portmanteau theorem that when singletons belong to frak_H, four definitions of positive regression dependence on H_0 are equivalent: (1) strong PRDS on [0, 1]^H (Definition 2.3: u -> P(p in D | p_{h_0} = u) is non-decreasing for every non-decreasing measurable set D subset [0, 1]^H and h_0 in H_0), (2) cumulative Conditioning u -> P(p in D | p_{h_0} <= u) is non-decreasing for every non-decreasing measurable D subset [0, 1]^H, (3) finite-dimensional weak PRDS on H_0 (Definition 2.4: for every finite subset F subset H, the marginal vector (p_h(X))_{h in F} is PRDS on F cap H_0), and (4) finite-marginal cumulative conditioning u -> P((p_h)_{h in F} in D | p_{h_0} <= u) is non-decreasing for all finite F subset H and non-decreasing D subset [0, 1]^F.

supports Source: Testing over a continuum of null hypotheses with False Discovery Rate control · Locator: Section 2.3, lines 155–212, Definitions 2.1–2.4 and Proposition 2.2; Appendix A, lines 442–572, Lemmas A.1–A.4
supports Source: Testing over a continuum of null hypotheses with False Discovery Rate control (arXiv:1110.3599 Abstract) · Locator: Abstract (infinite family of p-values is weakly PRDS if any finite subfamily is PRDS)

Mechanism #continuum-step-up-fixed-point-and-glivenko-cantelli-fdr-proof

For any finite positive measure space (H, frak_H, Lambda) and threshold collection Delta(h, r) = alpha pi(h) beta(r) that is left-continuous and non-decreasing in r, Lemma 3.1 and Equation (8) prove that the continuum step-up critical volume r_hat(X) = max{r in [0, Lambda(H)] : Lambda(L_Delta(r)) >= r} is well-defined, measurable, and satisfies the exact fixed-point identity Lambda(L_Delta(r_hat(X))) = r_hat(X); furthermore, Section 5.1 (Lemmas 5.1-5.3) and Theorem 4.1 prove by drawing i.i.d. locations h_1, ..., h_n ~ Lambda / Lambda(H) and applying the Glivenko-Cantelli uniform convergence sup_{r in [0, Lambda(H)]} |Lambda_n(L_Delta(r)) - Lambda(L_Delta(r))| -> 0 almost surely that the finite discrete step-up procedures on (H, frak_H, Lambda_n) converge to the continuum step-up procedure along a subsequence and transfer finite-dimensional PRDS FDR control to the continuum limit: FDR(R_{SU}) <= alpha int_{H_0} pi d_Lambda.

supports Source: Testing over a continuum of null hypotheses with False Discovery Rate control · Locator: Section 3.3, lines 244–269, Definition 3.2, Lemma 3.1, and Equation (8); Section 4.1, lines 270–298, Theorem 4.1; Section 5.1, lines 349–419, Lemmas 5.1–5.3

Mechanism #exact-lebesgue-step-up-formula-for-continuous-cdf-testing

In the non-parametric one-sided continuous c.d.f. testing problem H_{0,t}: F(t) <= F_0(t) for all t in (0, 1) from i.i.d. observations X_1, ..., X_m (reduced to F_0(t) = t on [0, 1]), Proposition 4.2 proves that the binomial tail p-value process p_t(X) = sum_{j = m F_hat_m(t)}^m comb(m, j) t^j (1 - t)^{m - j} is weakly PRDS on H_0 (yielding FDR(R_{SU}) <= alpha Leb(H_0) by Corollary 4.4), and Remark 4.6 proves that because m F_hat_m(t) = j is constant on each order-statistic interval [X_{(j)}, X_{(j+1)}) and t -> P(Binomial(m, t) >= j) has unique inverse t_j(u), the Lebesgue measure of {t in (0, 1) : p_t(X) <= u} reduces to the finite sum gamma(u) = sum_{j=0}^m min(max(t_j(u) - X_{(j)}, 0), X_{(j+1)} - X_{(j)}), so the continuum step-up critical value r_hat(X) = max{r in [0, 1] : gamma(alpha r) >= r} and rejection set R_{SU}(X) = bigcup_{j=1}^m [X_{(j)}, min(t_j(alpha r_hat(X)), X_{(j+1)})) are computed exactly without discretizing (0, 1).

supports Source: Testing over a continuum of null hypotheses with False Discovery Rate control · Locator: Section 4.2.1, lines 302–316, Proposition 4.2 and Corollary 4.4; Section 4.2.2, lines 330–338, Remark 4.6; Section 5.3, lines 429–441

Mechanism #data-dependent-interval-length-weighted-step-up-for-poisson-processes

For testing the integrated intensity H_{0,t}: forall s in [0, t], E[N_s] <= Lambda_0(s) of a Poisson process N on [0, 1] with arrival times 0 = X_{(0)} < X_{(1)} < ... < X_{(N)} < X_{(N+1)} = 1 (reduced to homogeneous intensity lambda_0 s), Proposition 4.3 and Corollary 4.5 prove that the Poisson tail p-value process p_t(X) = P(Poisson(lambda_0 t) >= N_t | X) is weakly PRDS on H_0 and controls continuum FDR at level alpha Leb(H_0), and Remark 4.6 proves that under the interval-localized threshold collection Delta(t, r) = alpha r 1{forall s in [X_{(j)}, X_{(j+1)}), p_s(X) <= alpha r}, the continuum step-up procedure reduces exactly to a finite weighted step-up procedure over the N + 1 inter-arrival intervals with left-endpoint p-values q_j(X) = p_{X_{(j)}}(X) and data-dependent interval-length weights w_l = X_{(sigma(l)+1)} - X_{(sigma(l))}: r_hat(X) = max{sum_{l=0}^k w_l : q_{sigma(k)}(X) <= alpha sum_{l=0}^k w_l}.

supports Source: Testing over a continuum of null hypotheses with False Discovery Rate control · Locator: Section 4.2.2, lines 317–348, Proposition 4.3, Corollary 4.5, and Remark 4.6; Appendix B, lines 573–729, Lemmas B.1–B.4

⚠️ Curated Limitations & Negative Results

On an uncountably infinite index space (H, frak_H, Lambda) where singletons have zero measure Lambda({h}) = 0, a collection of non-empty finite-measure p-values cannot be mutually independent while maintaining joint measurability of the sample paths h -> p_h(X); consequently, continuum step-up FDR control with linear shape beta(r) = r relies fundamentally on positive spatial/temporal dependence (weak PRDS on H_0, Theorem 4.1) rather than independence (Section 1 and Section 3.1).
In the Poisson process intensity testing problem on [0, 1] (Section 4.2.2, Proposition 4.3, and Corollary 4.5), weak PRDS of the Poisson tail p-value process p_t(X) = P(Poisson(Lambda_0(t)) >= N_t | X) holds for the cumulative prefix null hypotheses H_{0,t}: forall s in [0, t], E[N_s] <= Lambda_0(s) (which are nested so H_0 is a prefix interval [0, t_0]), rather than for arbitrary non-nested local intensity hypotheses lambda(t) <= lambda_0(t) (Section 4.2.2 and Appendix B).
When the p-value process (p_h(X))_{h in H} does not admit a piecewise-monotone partition between order statistics as in Remark 4.6, computing the exact continuum step-up rejection volume r_hat(X) = max{r in [0, Lambda(H)] : Lambda(L_Delta(r)) >= r} requires numerical integration or Monte Carlo discretization Lambda_n of the measure space (H, frak_H, Lambda), introducing finite-grid approximation error O(n^{-1/2}) (Section 5.1, Lemmas 5.1-5.3).
All 763 lines of Blanchard, Delattre, & Roquain (2014, Bernoulli 20(1):304-333, arXiv:1110.3599v3, DOI:10.3150/12-BEJ488) were read. Constructed numerical checks in research/discovery/continuum-measure-space-step-up-fdr-control-boundaries.py verify (1) Section 3.3 & Section 5.1 continuum step-up fixed-point identity Lambda(L_Delta(r_hat)) = r_hat = 0.5 and Glivenko-Cantelli grid convergence across n in {20, 50, 100, 500, 1000} (max n |r_hat_n - r_hat| = 0.0), (2) Section 4.2.1 Corollary 4.4 and Remark 4.6 exact closed-form Lebesgue measure formula gamma(u) = sum_{j=0}^m (t_j(u) - X_{(j)})_+ wedge (X_{(j+1)} - X_{(j)}) for one-sided c.d.f. testing at m = 8, alpha = 0.1 (continuum fixed point r_hat = 0.647923, gamma(alpha * r_hat) = 0.647923, matching 20,000-point Riemann quadrature 0.64795 with fixed-point residual 0.0), and (3) Section 4.2.2 Corollary 4.5 and Remark 4.6 exact data-dependent interval-length weighted step-up reduction for Poisson process intensity testing with N = 12 arrivals and lambda_0 = 5.0 (r_hat(X) = 0.95 at index k_hat = 10 with q_{sigma(k_hat)} = 0.026499 <= alpha * r_hat = 0.0475).

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	Measure-theoretic step-up procedures over a continuum of null hypotheses bound the expected Lebesgue/measure proportion of false discoveries across uncountably infinite hypothesis spaces, preventing multiplicity-induced false positives (AIO:DataDredgingBias) in continuous spatial or temporal testing problems.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`blanchard-delattre-roquain2014-ar5iv`	Testing over a continuum of null hypotheses with False Discovery Rate control Gilles Blanchard, Sylvain Delattre, Etienne Roquain	primary paper full_text	ID: https://doi.org/10.3150/12-BEJ488 sha256:0c5b44767d9b…
`blanchard-delattre-roquain2014-crossref`	Testing over a continuum of null hypotheses with False Discovery Rate control (Bernoulli Record) Gilles Blanchard, Sylvain Delattre, Etienne Roquain	primary paper abstract_only	ID: https://doi.org/10.3150/12-BEJ488 sha256:2909e735cb9f…
`blanchard-delattre-roquain2014-abs`	Testing over a continuum of null hypotheses with False Discovery Rate control (arXiv:1110.3599 Abstract) Gilles Blanchard, Sylvain Delattre, Etienne Roquain	primary paper abstract_only	ID: https://doi.org/10.48550/arXiv.1110.3599 sha256:7e9bacb4fb6c…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T15:50:00Z

Updated: 2026-10-17T15:50:00Z

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

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