Self-consistency and dependency control framework for false discovery rate control

⚠️ Curated Limitations & Negative Results

• While the Self-Consistency condition SC(alpha, pi, beta) holds automatically for all step-up, step-down, and generalized step-up-down procedures of order lambda (Section 3.1 and Theorem 4.5), the Dependency Control condition DC(beta) for the linear Simes shape function beta(r) = r requires either independence or Positive Regression Dependency on a Subset (PRDS on H_0) together with coordinatewise monotonicity (nonincreasingness) of |R(p)| in p; if p-values exhibit negative dependence or arbitrary correlation, linear beta(r) = r violates DC(beta) and can inflate FDR up to alpha * sum_{k=1}^m (1/k) (Section 3.2 and Section 4.1.2).
• Under completely unspecified/arbitrary p-value dependence, replacing beta(r) = r with a prior-integrated shape function beta_nu(r) = int_0^r x d_nu(x) guarantees DC(beta_nu) without any distributional or monotonicity assumptions on |R|, but requires selecting the prior probability measure nu on (0, infty) a priori: heavy-tailed harmonic priors nu({k}) propto 1/k (Benjamini-Yekutieli) lose power when many hypotheses are rejected, whereas uniform or linear priors (quadratic or cubic beta_nu) have strictly smaller thresholds at small rejection counts r = 1, 2 and can fail to reject isolated sparse signals (Section 4.1.2 and Figure 2).
• For adaptive step-up procedures Delta(h, r, p) = alpha_1 pi(h) G(p) beta(r), Lemma 4.3 requires the plug-in factor G(p) to be coordinatewise nondecreasing in p (so that p -> |R(p)| remains nonincreasing under PRDS) and adds the upper-tail overestimation probability P(G(p) > pi_0^{-1}) to the nominal bound alpha_1; standard one-stage Storey-type estimators G(p) violate this monotonicity under PRDS unless paired with a first-stage alpha_0-FWER bound as in Corollary 4.4 (Section 4.1.5 and Blanchard & Roquain 2009).
• All 839 lines of Blanchard & Roquain (2008, Electronic Journal of Statistics 2:963-992, arXiv:0802.1406v2, DOI:10.1214/08-EJS180) were read. Constructed numerical checks in research/discovery/self-consistency-and-dependency-control-fdr-framework-boundaries.py verify (1) Definition 3.1 & Section 3.1 order-free maximal self-consistent step-up fixed-point identity |L_Delta(r_hat)| = r_hat and subset maximality on a doubly weighted space with total volume Lambda(H) = 6.0 and integral Pi(H) = 1.0 (4 fixed-point and 4 exhaustive 2^m subset maximality checks, plus geometrically constrained contiguous subset [2, 3] subset R_SU = [0, 2, 3]), (2) Lemma 3.2(iii), Proposition 3.7, and Theorem 4.2 worst-case adversarial dependency control at m = 6, alpha = 0.1 (where linear Simes beta(r) = r has adversarial supremum ratio 2.4500000000000006 = gamma_6 and worst-case FDR = 0.24500000000000027 > 0.1, whereas Benjamini-Yekutieli harmonic, Sarkar uniform quadratic, linear-prior cubic, and Dirac delta_3 priors all satisfy adversarial supremum ratio = 1.0 and worst-case FDR = 0.1, with three-tier concentrated-signal rejection counts {'BY_harmonic': 0, 'Sarkar_uniform_quadratic': 5, 'linear_prior_cubic': 5} and {'BY_harmonic': 0, 'Sarkar_uniform_quadratic': 0, 'linear_prior_cubic': 5}), and (3) Theorem 4.6 log-convexity step-down bound across 300 (m, m_0) pairs (max endpoint ratio = 1.0000000000000002).