exchangeable largest crossing point asymptotic false discovery rate and global-null independence limit discontinuity
Mathematical Methods AIMech:ExchangeableLCPAsymptoticFDRAndIndependenceLimit DRAFT
Aliases: Finner-Dickhaus-Roters Exchangeable LCP Asymptotic FDR Framework, Dirac-Exchangeable Conditional Empirical CDF Largest Crossing Point (LCP) Method, Global-Null Double-Limit FDR Discontinuity Phi(-sqrt(-2*ln(alpha))) in Equicorrelated and Studentized Normal Models
An asymptotic multiple hypothesis testing analysis framework introduced by Helmut Finner, Thorsten Dickhaus, and Markus Roters (2007, Annals of Statistics 35(4):1432-1455, arXiv:0710.3171v1, DOI:10.1214/009053607000000046) that characterizes the limiting Expected Error Rate EER_infty(zeta) = lim_{n->infty} E[V_n/n] and limiting False Discovery Rate FDR_infty(zeta) = lim_{n->infty} E[V_n/(R_n vee 1)] of the linear step-up (LSU / Benjamini-Hochberg) procedure with Simes rejection curve r_alpha(t) = t/alpha under exchangeable test statistics T_i = g(X_i, Z) (i = 1, ..., n) generated by i.i.d. innovations X_1, ..., X_n and a common latent disturbance Z in Dirac-exchangeable configurations D-EX(zeta_n) with n_0/n -> zeta in (0, 1]. Conditioning on Z = z makes the p-values conditionally independent with limiting conditional empirical c.d.f. F_infty(t | z, zeta) = (1 - zeta) + zeta*(1 - W_X(g_1(W_T^{-1}(1 - t), z))) (Lemma 2.1, Equation (2.2)) and Largest Crossing Point (LCP) t(z | zeta) = sup{t in [alpha*(1 - zeta), alpha] : F_infty(t | z, zeta) = t/alpha} (Equation (2.3)). For zeta in (0, 1), V_n(z)/n -> t(z | zeta)/alpha - (1 - zeta) a.s. and V_n(z)/(R_n(z) vee 1) -> 1 - alpha*(1 - zeta)/t(z | zeta) a.s. (Theorems 2.1-2.2, Equations (2.4)-(2.9)). At the complete null zeta = 1 where t(z | 1) = 0 on E_0, Lemma 2.2 and Theorems 2.3-2.4 prove that FDR_infty(1 | z) = alpha*gamma(z) with right-hand derivative gamma(z) = lim_{t->0^+} F_infty(t | z)/t and FDR_infty(1) = P^Z(E_1) + alpha * int_{E_0} gamma(z) dP^Z(z) (Equations (2.10)-(2.15)). Applying this theory to equicorrelated normal variables D-EX-N(zeta_n) (Section 3, Theorems 3.1-3.3, Remark 3.1, Equations (3.1)-(3.6)) and jointly studentized normal variables D-EX-t(zeta_n) (Section 4, Lemma 4.1, Theorems 4.1-4.3, Equation (4.1)) proves that at zeta = 1, gamma(z) == 0 on E_0, and as the correlation rho -> 0^+ or degrees of freedom nu -> infty, lim_{rho->0^+} FDR_infty(1) = lim_{nu->infty} FDR_infty(1) = Phi(-sqrt(-2*ln(alpha))) < alpha for all alpha in (0, 1/2], demonstrating a strict non-commutativity of the sample-size limit n -> infty and the independence limit (rho -> 0^+ or nu -> infty) at the global null.
Identity Notes & Concept Scope
Distinct from all 442 pinned AIO concepts and all 600 previously curated AIMech concepts. While AIMech:AsymptoticallyOptimalRejectionCurveFDRControl (Finner, Dickhaus, & Roters 2009) derives the Dirac-uniform asymptotically optimal rejection curve f_alpha(t) = t / (t*(1 - alpha) + alpha) under independent p-values and regularized step-up-down/adjusted step-up procedures, and AIMech:ArchimedeanCopulaStepUpFDRBounds (Bodnar & Dickhaus 2014) derives finite-sample FDR bounds under Archimedean copula models using conditional independence given a latent frailty factor, neither record defines the Dirac-exchangeable model D-EX(zeta_n) = g(X_i, Z) with conditional empirical c.d.f. F_infty(t | z, zeta) and Largest Crossing Point t(z | zeta) (Lemma 2.1, Theorems 2.1-2.2, Equations (2.2)-(2.9)), the complete-null boundary derivative law FDR_infty(1) = P^Z(E_1) + alpha * int_{E_0} gamma(z) dP^Z(z) and shifted exponential MTP_2 invariance Example 2.3 (Lemma 2.2, Theorems 2.3-2.4, Equations (2.10)-(2.15)), the equicorrelated normal tangent-point jump discontinuity (t_1, t_2) and closed-form integrals for EER_infty(zeta) and FDR_infty(zeta) (Theorems 3.1-3.2, Remark 3.1, Equations (3.1)-(3.5)), or the global-null double-limit discontinuity lim_{rho->0^+} FDR_infty(1) = lim_{nu->infty} FDR_infty(1) = Phi(-sqrt(-2*ln(alpha))) < alpha in equicorrelated normal and jointly studentized t-models (Theorem 3.3, Equation (3.6), Lemma 4.1, Theorems 4.1-4.3, Equation (4.1)).
Claims & Grounded Evidence
4 Claims
Mechanism
#dirac-exchangeable-conditional-ecdf-and-largest-crossing-point-laws
Under the Dirac-exchangeable model D-EX(zeta_n) where T_i = g(X_i, Z) for i = 1, ..., n with n_0 = zeta_n*n true nulls (vartheta_i = 0) and n_1 = (1 - zeta_n)*n Dirac-zero false nulls (vartheta_i = infty), conditioning on Z = z renders the null p-values conditionally i.i.d. with limiting conditional empirical c.d.f. F_infty(t | z, zeta) = (1 - zeta) + zeta*(1 - W_X(g_1(W_T^{-1}(1 - t), z))) (Section 2.1, Lemma 2.1, page 1437, Equation (2.2)). Defining the Largest Crossing Point (LCP) of F_infty(. | z, zeta) with the Simes line r_alpha(t) = t/alpha by t(z | zeta) = sup{t in [alpha*(1 - zeta), alpha] : F_infty(t | z, zeta) = t/alpha} (Equation (2.3)), Theorem 2.1 (page 1438, Equations (2.4)-(2.7)) proves that for zeta in (0, 1) when P^Z(D_zeta) = 0 (where D_zeta is the set of tangent points without transversal crossing), lim_{n->infty} V_n(z)/n = t(z | zeta)/alpha - (1 - zeta) a.s. and lim_{n->infty} V_n(z)/(R_n(z) vee 1) = 1 - alpha*(1 - zeta)/t(z | zeta) a.s., yielding unconditional asymptotic limits EER_infty(zeta) = E^Z[t(Z | zeta)]/alpha - (1 - zeta) and FDR_infty(zeta) = 1 - alpha*(1 - zeta)*E^Z[1/t(Z | zeta)] along with exact limiting c.d.f.s G_{zeta,1} and G_{zeta,2} (Theorem 2.2, pages 1438-1439, Equations (2.8)-(2.9)).
Mechanism
#complete-null-zero-crossing-derivative-law-and-shifted-exponential-invariance
At the complete null zeta = 1 where F_infty(t | z, 1) = F_infty(t | z) can have Largest Crossing Point t(z | 1) = 0 on E_0 = {z : t(z | 1) = 0}, Lemma 2.2 (Section 2.2, page 1440, Equation (2.10)) proves that if n_0 i.i.d. variables have linear c.d.f. F_xi(t) = gamma*t on [0, alpha] with gamma <= 1/alpha and n_1 variables equal 0 a.s., then the exact finite-sample linear step-up FDR is E[V_n'/(R_n' vee 1)] = (n_0/n)*gamma*alpha. Example 2.3 (page 1441) shows that for shifted exponential variables T_i = X_i - Z with X_i, Z i.i.d. Exp(1), F_infty(t | z) = 2*exp(-z)*t on [0, exp(z)/2] has gamma(z) = 2*exp(-z), so for alpha -> 0 (or after truncating Z at ln(2*alpha)) the linear step-up procedure satisfies FDR_n(zeta_n) = zeta_n*alpha identically despite strong MTP_2 dependence. Generally, Theorems 2.3-2.4 (pages 1441-1443, Equations (2.12)-(2.15)) prove that if F_infty(t | z) has right-hand derivative at zero gamma(z) = lim_{t->0^+} F_infty(t | z)/t < 1/alpha on E_0, then FDR_infty(1 | z) = alpha*gamma(z) for z in E_0 and FDR_infty(1 | z) = 1 for z in E_1 = {z : t(z | 1) > 0}, giving the exact decomposition FDR_infty(1) = P^Z(E_1) + alpha * int_{E_0} gamma(z) dP^Z(z).
Mechanism
#equicorrelated-normal-tangent-point-jump-and-double-limit-discontinuity
In the equicorrelated normal model D-EX-N(zeta_n) where T_i = vartheta_i + sqrt(1 - rho)*X_i - sqrt(rho)*X_0 with rho in (0, 1) (Section 3, pages 1443-1446), F_infty(t | x_0) = 1 - Phi((Phi^{-1}(1 - t) + sqrt(rho)*x_0)/sqrt(1 - rho)) satisfies gamma(x_0) = 0 for all x_0 in R, and solving F_infty(t | x_0, zeta) = t/alpha for x_0 yields the unique crossing-point function x_0(t | zeta) = sqrt((1 - rho)/rho)*Phi^{-1}((1 - t/alpha)/zeta) - Phi^{-1}(1 - t)/sqrt(rho) (Equation (3.1)). Solving the simultaneous tangent point conditions d(u | x_0, zeta) = 0 and d_u(u | x_0, zeta) = 0 (Equations (3.3)-(3.4)) yields u_{1,2}(x_0) = -x_0/sqrt(rho) +- sqrt((1 - rho)/rho)*sqrt(x_0^2 - 2*ln(sqrt(1 - rho)/(alpha*zeta))) (Equation (3.5)), which defines a jump interval (t_1, t_2) on which the LCP jumps discontinuously at x_0(t_1 | zeta) = x_0(t_2 | zeta). Theorems 3.1-3.2 (pages 1444-1445) derive closed-form integrals for EER_infty(zeta) and FDR_infty(zeta), showing at the global null zeta = 1 that EER_infty(1) = int_{t_2/alpha}^1 Phi(x_0(alpha*t | 1)) dt and FDR_infty(1) = Phi(x_0(t_2 | 1)) <= Phi(-sqrt(2*ln(sqrt(1 - rho)/alpha))) (Remark 3.1). Theorem 3.3 (page 1445, Equation (3.6); proof on pages 1452-1453) proves that while lim_{rho->0^+} FDR_infty(zeta) = zeta*alpha for every fixed zeta in (0, 1), at the global null zeta = 1 the limit as correlation vanishes is lim_{rho->0^+} FDR_infty(1) = Phi(-sqrt(-2*ln(alpha))) < alpha for all alpha in (0, 1/2] (specifically 0.00718763 vs alpha = 0.05 at alpha = 0.05, a 6.9564x ratio).
Mechanism
#jointly-studentized-t-model-crossing-bounds-and-degrees-of-freedom-limit
In the jointly studentized normal model D-EX-t(zeta_n) where T_i = (vartheta_i + X_i)/S with nu*S^2 ~ chi_nu^2 independent of X_1, ..., X_n (Section 4, pages 1446-1450), F_infty(t | s) = 1 - Phi(s*t_{nu, 1-t}) has right-hand derivative gamma(s) = 0 for all s > 0, and solving F_infty(t | s, zeta) = t/alpha yields s(t | zeta) = Phi^{-1}((1 - t/alpha)/zeta) / t_{nu, 1-t}, which is positive only on the restricted crossing interval [t_u, t_o] = [alpha*(1 - zeta), alpha*(1 - zeta/2)] because t_{nu, 1-t} > 0 for t < 1/2 and Phi^{-1}((1 - t/alpha)/zeta) >= 0 iff t <= alpha*(1 - zeta/2) (pages 1446-1447). Theorems 4.1-4.2 (pages 1447-1448) express EER_infty(zeta) and FDR_infty(zeta) in terms of the chi-distribution c.d.f. F_{chi_nu}(sqrt(nu)*s(t | zeta)), with global-null values EER_infty(1) = int_{t_2/alpha}^{1/2} F_{chi_nu}(sqrt(nu)*s(alpha*t | 1)) dt and FDR_infty(1) = F_{chi_nu}(sqrt(nu)*s(t_2 | 1)). Using the asymptotic inflection-point scaling s_nu(x) = 1 - nu^{-1/2}*(-ln x)^{1/2} + o(nu^{-1/2}) and a(s, nu) = sqrt((nu + 1)/s^2 - nu) where a(s_nu(x), nu)^4/nu -> -4*ln(x) and a(s_nu(x), nu)^2*(1 - s_nu(x)^2) -> -4*ln(x) (Lemma 4.1, page 1448; proof on pages 1453-1454), Theorem 4.3 (page 1448, Equation (4.1)) proves via the Central Limit Theorem for (nu*S^2 - nu)/sqrt(2*nu) that lim_{nu->infty} FDR_infty(1) = Phi(-sqrt(-2*ln(alpha))), identical to the vanishing-correlation limit in the equicorrelated normal model.
⚠️ Curated Limitations & Negative Results
- In both the equicorrelated normal model D-EX-N(1) and the jointly studentized normal model D-EX-t(1), the sample-size limit n -> infty and the independence limit (rho -> 0^+ or nu -> infty) do not commute at the global null zeta = 1: taking n -> infty first and then rho -> 0^+ (or nu -> infty) yields Phi(-sqrt(-2*ln(alpha))) = 0.00718763 < alpha = 0.05 because the conditional empirical c.d.f. has zero right-hand derivative gamma(z) = 0 at t = 0, whereas taking rho -> 0^+ (or nu -> infty) at fixed n yields the independent global-null FDR equal to alpha (Theorems 3.3 and 4.3, pages 1445-1450).
- While the asymptotic FDR in D-EX-N(zeta) and D-EX-t(zeta) is bounded above by zeta*alpha, the unconditional Expected Error Rate EER_infty(zeta) can substantially exceed the independent EER limit alpha*zeta*(1 - zeta)/(1 - alpha*zeta) under moderate correlation rho or small degrees of freedom nu, and the conditional false discovery proportion V_n(z)/(R_n(z) vee 1) exhibits high variance and heavy clustering near 0 and 1 across realizations of the latent factor Z (Figures 2 and 4, Section 5 Concluding Remarks, pages 1445-1451).
- In the prose following Theorem 3.3 on page 1446 (lines 6-7) and following Theorem 4.3 on page 1448 (lines 32-33), the paper writes that as rho -> 0 (or nu -> infty), EER_infty(zeta) tends to alpha*(1 - zeta)/(1 - alpha*zeta) for zeta in (0, 1); that prose expression omits the factor zeta from the true independent limit EER_infty(zeta) = t^*/alpha - (1 - zeta) = zeta*t^* = alpha*zeta*(1 - zeta)/(1 - alpha*zeta) (where t^* = alpha*(1 - zeta)/(1 - alpha*zeta) is the independent crossing point, as correctly reflected by the integral in Theorem 3.1 and Figure 2 on page 1445 where EER_infty(0.5) -> 0.01282051 at alpha = 0.05 rather than 0.02564103).
- All 24 pages of the full-text PDF of Finner, Dickhaus, & Roters (2007, Annals of Statistics 35(4):1432-1455, arXiv:0710.3171v1, DOI:10.1214/009053607000000046) were read. Constructed numerical checks in research/discovery/exchangeable-lcp-asymptotic-fdr-and-independence-limit-boundaries.py verify (1) Lemma 2.2 and Example 2.3 exact conditional linear step-up FDR_n(zeta_n | z) = (n_0/n)*gamma(z)*alpha = 0.11982106 via Bolshev/Steck recursion and exact unconditional invariance FDR_n(zeta_n) = (n_0/n)*alpha = 0.13333333 for shifted exponential variables T_i = X_i - Z, (2) Section 3 Equations (3.1)-(3.5), Theorem 3.2, and Remark 3.1 tangent-point solutions and upper bounds across rho in [0.0001, 0.5], (3) Theorem 3.3 exact algebraic Mills-ratio exponent identity exp((w^2 - u^2)/2) == delta and global-null double-limit discontinuity Phi(-sqrt(-2*ln(0.05))) = 0.00718763 vs alpha = 0.05, (4) Theorem 3.1 numerical integration at zeta = 0.5, alpha = 0.05 confirming convergence of EER_infty(0.5) to 0.01282051 and FDR_infty(0.5) to 0.025 as rho -> 0^+, and (5) Lemma 4.1 and Theorem 4.3 studentized D-EX-t(1) inflection-point scaling a(s_nu(alpha), nu)^4/nu -> -4*ln(alpha) = 11.982929.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:DataDredgingBias
Data Dredging Bias
|
related_match | Exchangeable Largest Crossing Point (LCP) asymptotic FDR theory quantifies the exact limiting false discovery rate and expected error rate of linear step-up multiple testing under equicorrelated, studentized, and general exchangeable dependence structures, directly addressing multiplicity-induced false positives (AIO:DataDredgingBias). |
Evidence Sources & Primary Literature
3 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
finner-dickhaus-roters2007-pdf
|
Helmut Finner, Thorsten Dickhaus, Markus Roters
|
primary paper
full_text
|
ID: https://doi.org/10.1214/009053607000000046
sha256:4ccb7542eb74…
|
finner-dickhaus-roters2007-crossref
|
Helmut Finner, Thorsten Dickhaus, Markus Roters
|
primary paper
abstract_only
|
ID: https://doi.org/10.1214/009053607000000046
sha256:e64def78de01…
|
finner-dickhaus-roters2007-abs
|
Helmut Finner, Thorsten Dickhaus, Markus Roters
|
primary paper
abstract_only
|
ID: https://doi.org/10.48550/arXiv.0710.3171
sha256:734f9e1d24f8…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini 3
Created: 2026-10-17T16:35:00Z
Updated: 2026-10-17T16:35:00Z
Source File:
data/curated/ExchangeableLCPAsymptoticFDRAndIndependenceLimit.yaml
· Schema:
curated_concept.yaml
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