{"id":"bee5fb5c-923b-4acc-8c18-d1fe07682afb","arxiv_id":"1908.02193","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For equicorrelated normal test statistics with common correlation ρ, the Bonferroni FWER is asymptotically bounded above by α(1-ρ).","lead":"This paper derives an asymptotic upper bound on the family-wise error rate (FWER) of the Bonferroni procedure when the test statistics are equicorrelated standard normals. The bound, α(1-ρ), says positive correlation makes Bonferroni even more conservative than the nominal level.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof's second-derivative analysis uses an incorrect G; with the correct derivative the integrand acquires extra 1/ρ and 1/√ρ factors, so the localization argument is not established near ρ=0.","rationale":"The paper has a clear and plausible goal: to show asymptotic concavity of H(ρ) for equicorrelated normal maxima, giving FWER ≤ α(1-ρ). The endpoint values are correct, the simulation results are broadly supportive, and the claimed bound is qualitatively reasonable. The reader's conditional verdict is therefore appropriate. However, the central proof has a concrete, checkable defect: the derivative G is miscomputed. With the correct G, the H'' integrand is more singular near ρ=0, and the proof's vanishing/localization lemmas are only sketched and are not shown to be uniform in ρ. The incorrect T(ρ) in §4.1 compounds this. These are technical proof failures rather than demonstrated counterexamples; the theorem may still be true, but as written the asymptotic convexity is not established for the small-ρ regime where the bound has content. I recommend keeping the conditional verdict: the paper should be accepted only after a corrected re-derivation of the derivative formulas and of Lemmas 6.1 and 6.2, preferably with explicit uniformity in ρ over some interval (0,ρ_0).","tokens_in":8430,"tokens_out":11116,"duration_ms":117814,"concrete_test":"Symbolically differentiate d=(c+√ρ Z)/√(1-ρ), re-derive H'(ρ) and H''(ρ), and verify whether equations (1)-(2) hold with G_old=(c+Z√ρ)/(2(1-ρ)^{3/2}); then recompute the proof's estimates with the corrected G. For a direct numerical check, evaluate the true H''(ρ) by high-precision Gauss-Hermite quadrature for n=10^4, α=0.05, ρ=0.001, 0.005, 0.01, 0.05, and 0.1; if any value is positive, Theorem 3.1 is false rather than merely unproved. Separately, re-run the Table 1 simulations with n=10^5, α=0.01, and ρ=0.001 through 0.05; a violation of FWER ≤ α(1-ρ) in that small-ρ window would confirm the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.1 and Corollary 3.1.1 rest on the asymptotic sign of H''(ρ). The derivative G defined before equation (1) is miscomputed: from d=(c+√ρ Z)/√(1-ρ), the correct derivative is G = (c + Z/√ρ)/(2(1-ρ)^{3/2}), not (c+Z√ρ)/(2(1-ρ)^{3/2}). This is not harmless. It changes the ρ-singularity of every term in the H'' integrand: the first term is approximately G^2 (extra 1/ρ), the second term bG has b∼1/ρ and G∼1/√ρ (extra 1/ρ^{3/2}), and the second/third-term vanishing estimates in Lemma 6.1 and the localization in Step 2 are not uniform as ρ→0. The same subsection contains a related constant error: solving z0=(√(1-ρ)d(z0)-c)/√ρ with d(z0)≈c gives z0≈-c√ρ/(1+√(1-ρ)), so T(ρ) should be √ρ/(1+√(1-ρ)), not 1/(1+√(1-ρ)). This weakens the claimed control of sup_{z≤z0} dG^2φ(z) for small positive ρ. Lemma 6.1 explicitly proves only the third term and says the other follows similarly; Lemma 6.2 is a sketch. Since the asymptotic convexity of FWER, and hence the α(1-ρ) bound, depends on these estimates, the proof as written does not establish the claim in the small-ρ regime, which is exactly the regime where the bound is nontrivial.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the family-wise error rate (FWER) of a Bonferroni-type procedure when the n test statistics are equicorrelated standard normal variables. The central object is H(ρ) = P(max_i X_i ≤ c), and the authors claim that as n → ∞ with n α_n → α, H is concave in ρ ∈ [0,1], so FWER is convex and lies below the chord joining the independence and perfect-correlation endpoints. For Bonferroni's choice α_n = α/n this yields the asymptotic bound FWER ≤ α(1−ρ). The proof represents H as an expectation over a common normal factor, differentiates twice under the integral, and then analyzes the sign of H'' by splitting the integration region according to α_1 = Φ(−d) relative to 1/n. Simulations in Table 1 are reported to support the bound for n = 10000 across several α and ρ.","tokens_in":8713,"tokens_out":10101,"duration_ms":96600,"significance":"If the claim is correct, the paper provides a very clean, parameter-free asymptotic bound that quantifies how positive equicorrelation makes Bonferroni's procedure more conservative, and it identifies a simple correction factor. The bound is falsifiable and the simulation evidence in Section 7 is broadly supportive, with all reported estimates below α(1−ρ). The representation of H as an expectation and the reduction of the problem to the sign of H'' are natural and potentially useful. However, the proof as written contains a miscomputed derivative, an incorrect constant in the location of z0, and asymptotic lemmas that are only sketched; these issues directly affect the claimed concavity in the small-ρ regime, which is essential for the final bound. The result is plausible, but the manuscript does not currently establish it rigorously.","major_comments":[{"comment":"The derivative G in Eq. (1) is incorrect. With d = (c + √ρ Z)/√(1−ρ), the correct derivative is ∂d/∂ρ = (c + Z/√ρ)/(2(1−ρ)^{3/2}), not (c + Z√ρ)/(2(1−ρ)^{3/2}) as stated. This is not a harmless typo: under the correct G, the first term in H'' acquires an extra 1/ρ factor, the term bG acquires an extra 1/ρ^{3/2} factor (since b ∼ 1/ρ), and the estimates in Lemmas 6.1 and 6.2 acquire additional singularities as ρ → 0. The proof's localization argument relies on bounds such as sup_{z≤z0} dG²φ(z) → 0 as n → ∞; with the correct G this sup is not bounded uniformly in ρ, and the claim that the integrals over {α1 ≥ 1/n} vanish is not established in the small-ρ regime. Since Theorem 3.1 asserts concavity on all of [0,1], and the small-ρ regime is exactly where the bound is nontrivial, the central proof does not currently go through.","section":"Eq. (1) and Section 3.1"},{"comment":"The derivation after c − d(z0) → 0 concludes that z0 + cT(ρ) → 0 with T(ρ) = 1/(1 + √(1−ρ)). Solving z0 = (√(1−ρ) d(z0) − c)/√ρ with d(z0) ≈ c gives z0 ≈ −c√ρ/(1 + √(1−ρ)), so T(ρ) should be √ρ/(1 + √(1−ρ)). This quantity tends to 0 as ρ → 0. The subsequent estimates in Lemma 6.1 (e.g., c e^{−z0²} → 0) require |z0| to grow like a constant times c; for small ρ this fails. Thus the control of the middle region in Lemma 6.1 is not uniform in ρ, and the localization argument on which Step 2 of the main proof depends is not valid as ρ → 0.","section":"Section 4.1, location of z0"},{"comment":"The definition of a in Eq. (2) is inconsistent with the approximation used in the proof. The paper defines a = (n−1)φ(d) − dΦ(−d), but Lemma 6.2 uses the identity |a − d(nα1 − 1)| = (n−1)|φ(d) − dΦ(−d)|. This identity holds only if a = (n−1)φ(d) − dΦ(d), i.e., the argument of Φ should be d, not −d. As written, a − d(nα1 − 1) equals (n−1)φ(d) − dα1 − dnα1 + d, not (n−1)(φ(d) − dα1). Since Lemma 6.2 is the step that replaces a by d(nα1 − 1) and thereby determines the sign of the dominant term, this inconsistency is load-bearing. Correcting it to Φ(d) is necessary, and the proof of Lemma 6.2 would need to be redone with the corrected formula.","section":"Eq. (2) and Lemma 6.2"},{"comment":"The asymptotic localization is not established with sufficient rigor. Lemma 6.1 explicitly proves only the third term and states that the second 'follows similarly'; Lemma 6.2 is a sketch that says 'an idea similar to the proof of lemma 6.1 will tell us' and considers only the region {1/(n(log n)^3) ≤ α1 ≤ 8 log n/n}. In view of the ρ-singularities introduced by the correct derivative G, those 'similar' arguments are not routine and require explicit uniform bounds in ρ and n. Moreover, the final conclusion of Theorem 3.1 requires H_n''(ρ) ≤ 0 for all ρ ∈ [0,1] for large n; a pointwise statement for each fixed ρ would not justify the chord bound in Corollary 3.1.1. The manuscript needs to state and prove uniform versions of the estimates, or restrict the theorem to a setting where uniformity is not needed.","section":"Section 4 and Appendix, Lemmas 6.1 and 6.2"}],"minor_comments":[{"comment":"The symbol c is used both for the critical value and then again inside formula (2); this is confusing. Please distinguish the critical value, e.g., c_n, from other uses.","section":"Section 4, notation"},{"comment":"The text writes 'n ∼ α√(2π c) e^{c²/2}'; the correct relation from α_n ∼ φ(c)/c is n ∼ α √(2π) c e^{c²/2}. Please fix the typo.","section":"Section 4, asymptotic relation"},{"comment":"The inequality φ(d) ≤ d³ α1 for large d is weaker than the standard φ(d) ∼ d α1; using the sharper asymptotic would simplify the proof and avoid unnecessary logarithmic factors. Please clarify.","section":"Lemma 6.1"},{"comment":"The table header is visually confusing: the row label 'ρ α' followed by the column labels 0.01, 0.05, ... without separating the α values from the ρ values. Please reformat the table so the two factors are clear.","section":"Table 1"},{"comment":"The paper says the Tong (2014) bound is 'not a bounded quantity as n gets larger'; please clarify what is meant, since a probability is always in [0,1]. Presumably the comparison is about the sharpness of the upper bound as n grows, not its lack of finite value.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript addresses a real and interesting question, and the claimed bound α(1−ρ) is attractive and well supported by the simulations. However, the proof as written contains a miscomputed derivative and an incorrect constant in the derivation of z0, and the asymptotic lemmas are not rigorous enough to establish uniformity in ρ. These are not merely presentation issues; they affect the validity of the central concavity claim in the small-ρ regime. I believe the result is likely recoverable, but the authors must supply a correct and complete proof of the localization with the proper G, including uniform estimates as ρ → 0, or else restrict the theorem to a setting where the proof works. I would not reject outright because the core idea is plausible and the numerical evidence is consistent, but the current manuscript cannot be accepted as a proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core result—FWER ≤ α(1−ρ) for Bonferroni under equicorrelated normal tests—is a nice, memorable bound and likely true. The one-factor representation H(ρ)=E[Φ^n((c+√ρ Z)/√(1−ρ))] is clean, and the idea of proving concavity of H via the second derivative is sensible. The simulations back the bound, and the paper is self-contained with no fitted parameters and no self-citation problem. Credit where it is due: the specific asymptotic bound and the convexity argument are new relative to the cited literature, and the writing is understandable despite the typos.\n\nBut the proof has a concrete error that the reader's report underweights. The derivative G defined before equation (1) is miscomputed. The correct derivative of d=(c+√ρ Z)/√(1−ρ) is (c+Z/√ρ)/(2(1−ρ)^{3/2}), not (c+Z√ρ)/(2(1−ρ)^{3/2}). This is not a harmless typo: it changes the ρ-singularity of the integrand. The term dG^2 picks up a factor 1/ρ, and the claimed sup_{z≤z0} dG^2 φ(z) → 0 is not established uniformly in ρ. The same pass gets T(ρ) wrong: it should be √ρ/(1+√(1−ρ)), not 1/(1+√(1−ρ)). For a fixed ρ>0 these errors may be repairable—the sup still goes to zero like c^3 e^{-c^2 ρ/4}—but as written the localization argument does not cover the small-ρ regime, which is exactly where the bound is most interesting.\n\nLemmas 6.1 and 6.2 are also sketches. Lemma 6.1 explicitly proves only the third term and says the second follows similarly; lemma 6.2 is a similar hand-wave. The central step of the proof depends on these lemmas. The sign typo in the definition of a is minor; the asymptotic a ~ d(nα1−1) is correct.\n\nWho is this for? People working on multiple testing under dependence, and anyone wanting a simple benchmark for how positive correlation reduces FWER. It does not reshape the field, but it is a clean addition if the proof can be made rigorous.\n\nMy recommendation: do not desk-reject. Send it to a serious referee, but expect a major revision. The referee should redo the second derivative computation and complete the two lemmas. If the authors can do that, the paper is publishable. As it stands, it is a plausible conjecture with supporting simulations and an incomplete proof.","headline":"A simple and probably true bound for FWER under equicorrelated normality, but the proof has a real derivative error and sketched lemmas; worth a serious referee, not acceptance as-is.","tokens_in":9302,"tokens_out":12280,"would_cite":false,"duration_ms":113279,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F03","62E20","62H20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, for equicorrelated normal test statistics, the Bonferroni family-wise error rate is asymptotically bounded by alpha(1-rho).","keywords":["family-wise error rate","Bonferroni procedure","equicorrelated normal","multiple testing","convexity","asymptotic bound","correlation","simultaneous hypothesis testing"],"falsifier":"Numerically integrate H(rho) = E[Phi^n((c + $\\sqrt$(rho)Z)/$\\sqrt$(1-rho))] for, say, n=$10^{4}$ and $\\alpha$=0.05 over a fine grid of rho in [0,1], computing H''(rho) by quadrature; one interval with H''(rho) > 0 at large n would refute the claimed asymptotic concavity, and hence the $\\alpha$(1-rho) bound.","tokens_in":8168,"feed_emoji":"📉","tokens_out":6202,"duration_ms":60962,"temperature":0.7,"pith_summary":"The paper sets out to show that when normally distributed test statistics are positively equicorrelated, the family-wise error rate of a Bonferroni multiple-testing procedure shrinks as the correlation grows, and is asymptotically bounded by alpha(1-rho) rather than alpha. If true, this means the classical Bonferroni correction is conservative under positive dependence, and that a correlation-aware correction is needed to avoid over-correction. The argument runs through the probability H(rho) that no null is rejected: showing H is asymptotically concave in rho makes FWER convex, so it is controlled by the straight line between the independence and perfect-dependence endpoints.","feed_headline":"Correlated tests cut Bonferroni's error bound to alpha(1-rho)","feed_subtitle":"Large-n proof that family-wise error is convex in correlation, making the classical bound conservative.","key_machinery":"The central object is H(rho), the probability that all n standardized test statistics stay below the common cutoff c. The paper rewrites equicorrelated normals as X_i = $\\theta$ + Z_i, where $\\theta$ is a common N(0,rho) component independent of the i.i.d. N(0,1-rho) noise Z_i, so that H(rho) = E[Phi^n((c + $\\sqrt$(rho)Z)/$\\sqrt$(1-rho))]. Differentiating twice under the integral, it shows the terms in H''(rho) vanish except on the tail region alpha_1 < 1/n, where alpha_1 = Phi(-d) is the single-test tail probability, and that the remaining term is non-positive. This asymptotic concavity of H is the load-bearing identity: it converts the two endpoint values into a linear upper bound on FWER.","core_discovery":"For n null hypotheses X_i ~ N(0,1) with common correlation rho, testing each at level alpha_n with n*alpha_n -> alpha, the paper claims that H(rho) = P(max_i X_i <= c) is asymptotically concave in rho. Consequently FWER = 1 - H is asymptotically convex in rho and lies below the chord joining the independence endpoint (rho=0, FWER ~ n*alpha_n) to the perfect-dependence endpoint (rho=1, FWER=alpha_n). With Bonferroni's choice alpha_n = alpha/n, this gives the asymptotic bound FWER(rho) <= alpha(1-rho), showing that positive correlation reduces the effective family-wise error rate below the nominal level.","pith_inferences":["A direct extension the authors leave implicit is a correlation-correction recipe: to hold FWER near alpha under positive correlation, a Bonferroni test could inflate its marginal level to roughly alpha/(1-rho), although the paper does not propose such a procedure.","If the asymptotic concavity could be established non-asymptotically, the same chord argument would give a finite-sample bound; the natural numerical check is whether H''(rho) <= 0 already holds at moderate n.","The proof's localization to the tail region alpha_1 < 1/n suggests the result is driven by rare single-test events, so a similar bound might hold for other exchangeable distributions, but any extension would need a new integral representation because the Gaussian identity is used explicitly."],"forward_implications":["For Bonferroni's rule alpha_n = alpha/n, the family-wise error rate is asymptotically no larger than alpha(1-rho), so positive correlation makes the procedure more conservative than the nominal level suggests.","FWER as a function of rho is asymptotically convex, so its worst case over rho in [0,1] occurs at rho=0, meaning independence is the most error-prone configuration.","The bound remains bounded and meaningful as n grows, unlike the distribution-free bound in Tong (2014), which becomes unbounded as the number of hypotheses increases.","In the parallel-system lifetime model, the result gives a lower bound on the c.d.f. of the failure time when component lifetimes are exchangeable normal variables.","The result indicates that Bonferroni-type procedures under positive correlation need a correlation correction to avoid being overly conservative."],"supporting_citations":[{"why":"Supplies the distribution-free equicorrelated FWER bound that the paper identifies as unbounded in n and replaces with the normal-specific alpha(1-rho) asymptotic bound.","marker":"Tong (2014)"},{"why":"Documents the correlation penalty in large-scale significance testing and motivates a bound that depends explicitly on the correlation rho.","marker":"Efron (2010)"},{"why":"Provides the skew-normal representation of the maximum of exchangeable normal random vectors, the setting where the paper's bound becomes a lower bound on the parallel-system failure-time c.d.f.","marker":"Loperfido et al. (2007)"}],"fun_headline_variants":["Correlated normals shrink Bonferroni's FWER bound to alpha(1-rho)","Equicorrelated normal tests: FWER bounded by alpha(1-rho)","Family-wise error for correlated normals is at most alpha(1-rho)","Positive correlation implies FWER <= alpha(1-rho) in normal tests"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the assumption that, for large n, the second derivative of H(rho) is controlled by the tail region alpha_1 < 1/n and that this localization is uniform in rho; if that fails, the convexity of FWER and the alpha(1-rho) bound are not established.","fun_headline_variants_meta":{"raw":{"variants":["Correlated normals shrink Bonferroni's FWER bound to alpha(1-rho)","Equicorrelated normal tests: FWER bounded by alpha(1-rho)","Family-wise error for correlated normals is at most alpha(1-rho)","Positive correlation implies FWER <= alpha(1-rho) in normal tests"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001379,"raw_usage":{"total_tokens":5575,"prompt_tokens":927,"completion_tokens":4648,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":4561}},"tokens_in":543,"tokens_out":4648,"duration_ms":46528,"temperature":1.0,"reasoning_tokens":4561,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:54:12.634444+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically integrate H(rho) = E[Phi^n((c + $\\sqrt$(rho)Z)/$\\sqrt$(1-rho))] for, say, n=$10^{4}$ and $\\alpha$=0.05 over a fine grid of rho in [0,1], computing H''(rho) by quadrature; one interval with H''(rho) > 0 at large n would refute the claimed asymptotic concavity, and hence the $\\alpha$(1-rho) bound.","supporting_citations":[],"review_version":1}