{"id":"2ded8faa-0893-479c-be89-35010cba42a7","arxiv_id":"2411.12119","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For equicorrelated test statistics with light-tailed noise, the Bonferroni family-wise error rate goes to zero as the number of hypotheses grows, generalizing a known normal result.","lead":"This paper extends results on how the Bonferroni family-wise error rate behaves under positively correlated test statistics from normal distributions to a wider class of light-tailed distributions. It proves the error rate tends to zero as the number of hypotheses grows, and attempts to extend this to elliptically contoured distributions and step-down procedures.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1's proof collapses on a simple correlation matrix where liminf rho_ij = 1 but M_n is empty; the claimed extension to general positive correlation is unproven.","rationale":"The reader's weakest_assumption was the restrictive superexponential tail condition (ii) in Theorem 3.2, which limits the claimed 'wide class' of distributions but does not invalidate the theorem as stated. I find a more load-bearing problem in the proof of Theorem 4.1's extension to general positive correlation: the finiteness of n - |M_n| is asserted without justification and is false for a natural correlation structure satisfying the hypotheses. This is not a matter of applicability but of proof validity for a central result advertised in the abstract. The reader's rationale did list this as one of the gaps, so there is partial agreement, but the primary weakest_assumption differs. The equicorrelated zero-limit result (Theorem 3.3) appears mathematically sound, so I do not recommend rejection; the existing CONDITIONAL verdict is appropriate, requiring a corrected proof or a clarified definition of liminf for Theorem 4.1.","tokens_in":13224,"tokens_out":18514,"duration_ms":207805,"concrete_test":"Construct the infinite correlation matrix with rho_{1j} = 1/2 for j >= 2 and rho_{ij} = 1 for all i,j >= 2. Verify that liminf_{i,j -> infinity} rho_{ij} = 1 while M_n = empty for every n, so n - |M_n| = n. Then re-run the proof of Theorem 4.1 on this matrix: the claimed upper bound becomes F W E R_Bon(|M_n|, alpha, delta) + (n - |M_n|) * alpha/n with an empty first term and a second term equal to alpha, so the conclusion does not follow. A repair attempt can be tested by taking M_n = {2, ..., n} instead and checking whether the same argument yields limsup <= 0, but the current proof needs this replacement explicitly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main advertised advance beyond the equicorrelated case is the extension to general positive correlation, Theorem 4.1. Its proof defines M_n = {i : rho_ij >= delta for all j != i} and asserts that liminf_{i,j} rho_ij = delta > 0 implies n - |M_n| is finite. This assertion is false under the standard infinite-array reading of liminf_{i,j -> infinity}. Take rho_{1j} = 1/2 for all j >= 2 and rho_{ij} = 1 for all i,j >= 2. Then every pair with both indices tending to infinity has correlation 1, so liminf = 1 > 0. Yet for every n, M_n is empty: index i >= 2 has j = 1 with correlation 1/2 < 1, and index 1 has j = 2 with correlation 1/2 < 1. Hence n - |M_n| = n, unbounded, and the proof's term (n - |M_n|) * alpha/n equals alpha, not 0. The reduction to Theorem 3.3 via DasGupta's inequality therefore fails. The theorem may be salvageable by selecting a large subset with pairwise correlations at least some delta' > 0, but the proof as written does not establish it. This is a concrete, locatable gap in a central advertised result, not merely a restrictive assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the asymptotic family-wise error rate (FWER) of Bonferroni and step-down multiple testing procedures for correlated test statistics under non-normal distributions. In the equicorrelated factor model X_i = sqrt(1-ρ)Z_i + sqrt(ρ)U, the paper proves an upper bound for the Bonferroni FWER (Theorem 3.1), derives a zero limit under a superexponential tail condition on the density f of the idiosyncratic components (Theorems 3.2–3.3), and attempts to construct a Pareto example with a strictly positive FWER limit (Section 3.2, Theorem 3.4). It then claims an extension to general positively correlated elliptically contoured distributions (Theorem 4.1) and to step-down procedures (Theorem 5.3).","tokens_in":13430,"tokens_out":29939,"duration_ms":294413,"significance":"If the central zero-limit theorem is correct, it is a genuine generalization of the known equicorrelated-normal results to a class of light-tailed distributions, and the proof in Section 3 is largely self-contained, built on conditioning and standard inequalities rather than on fitted parameters. The claimed extensions to general positive correlation and to all step-down procedures would considerably broaden the applicability of the results. However, as submitted, the positive-limit example and the two advertised extensions contain load-bearing gaps. The core zero-limit result for the equicorrelated factor model appears defensible after correcting typos, so the paper has real potential, but it is not yet in publishable form.","major_comments":[{"comment":"The proof of the positive-limit example is incorrect. The displayed identity F^n( sqrt(n/(γ(1-ρ))) ) = (1 - d/n^{1+δ/2})^n is followed by the lower bound (1 - d/n^{1+δ/2})^{n^{1+δ/2}} -> e^{-d}. But the left-hand side actually tends to 1, because n/n^{1+δ/2} -> 0; a lower bound by e^{-d} does not establish that the limit of F^n at this threshold is below 1, which is what the positive-FWER conclusion requires. Moreover, condition (ii) of Theorem 3.4 cannot hold for any F with finite second moment, which the model assumes: for c_n = sqrt(n/(γ(1-ρ))), finite variance implies c_n^2(1-F(c_n)) -> 0, hence n(1-F(c_n)) -> 0 and F^n(c_n) -> 1. The claimed Pareto phenomenon is in fact true, but it must be shown using the actual quantile c_Bon: one obtains F^n(c_Bon/sqrt(1-ρ)) = (1 - α(1-ρ)^{1+δ/2}/n)^n -> exp(-α(1-ρ)^{1+δ/2}) < 1. This section therefore needs to be rewritten with a correct calculation and a corrected sufficient condition.","section":"Section 3.2, Theorem 3.4 and the Pareto example"},{"comment":"The assertion that 'Since lim inf ρ_ij = δ > 0, we also have n - |M_n| is finite' is false under the standard double-array reading of liminf. For example, set ρ_{1j} = 1/2 for all j ≥ 2 and ρ_{ij} = 1 for all i,j ≥ 2. Then every pair with both indices tending to infinity has correlation 1, so the liminf is 1, but for every n the set M_n = {i : ρ_{ij} ≥ 1 for all j ≠ i} is empty; hence n - |M_n| = n is unbounded. The reduction to Theorem 3.3 therefore fails at a load-bearing step. The proof needs a different construction of a large subset with all pairwise correlations bounded below by some δ' > 0, rather than the set M_n as currently defined.","section":"Section 4, proof of Theorem 4.1"},{"comment":"The statements assume an elliptically contoured joint density but formulate the sufficient conditions in terms of densities f and g from the factor model X_i = sqrt(1-ρ)Z_i + sqrt(ρ)U of Section 2.1, and the proofs invoke Theorem 3.3, which is proved only for that factor model. An equicorrelated elliptical distribution need not admit the representation with independent Z_i and U outside the normal case, so the meaning of f and g in Theorems 4.1 and 5.3 is not defined and the reduction to Theorem 3.3 is incomplete. The authors need either to prove the zero-limit result directly for equicorrelated elliptical laws or to state explicit conditions on the radial density or on the marginal distribution.","section":"Section 4, Theorems 4.1 and 5.3"},{"comment":"The proof uses the inequality P_{Σ_n}(X_(n) ≥ c_Bon) ≤ P_{Γ_n(δ)}(X_(n) ≥ c_Bon), which requires ρ_{ij} ≥ δ for all i,j. The theorem only assumes liminf ρ_{ij} = δ, so the same issue as in Theorem 4.1 arises. In addition, the final step asserts that E_U[F^n((c_Bon - sqrt(ρ)U - μ*)/sqrt(1-ρ))] -> 1 with the explanation that the proof is exactly similar to Theorem 2 of Dey and Bhandari (2023b), but no argument is supplied for why the fixed shift μ* is compatible with condition (ii). Since this is the step that transfers the equicorrelated zero-limit to arbitrary configurations with bounded means, the details need to be written out.","section":"Section 5, proof of Theorem 5.3"}],"minor_comments":[{"comment":"The statements read 'lim_{n→0}' in several places; this should be 'lim_{n→∞}'.","section":"Theorems 3.2, 3.3, 3.4"},{"comment":"In the displayed chain of inequalities, the integration upper limit 'c_Bon · (1-α)/sqrt(ρ)' should be 'c_Bon · (1-d)/sqrt(ρ)', and the expression 'c_Bon · 1-d/sqrt(ρ)' is missing parentheses; it should be 'c_Bon(1-d)/sqrt(ρ)'.","section":"Proof of Theorem 3.1"},{"comment":"The constant d in the Pareto calculation is not defined correctly: the expression (1 - d/n^{1+δ/2})^n would require d = η^{2+δ}(γ(1-ρ))^{1+δ/2}, not d = η^{1+δ/2}. This is part of the larger algebraic error described in the major comments.","section":"Section 3.2"},{"comment":"The claim that condition (ii) is satisfied by 'a wide class of distributions from the exponential family' is not demonstrated and is false for the standard natural exponential family on R+, where f(x)/f(x-b) is typically a constant. The authors should state precisely which parametric families satisfy the condition.","section":"Theorem 3.2, condition (ii)"},{"comment":"The notation liminf ρ_{ij} for a triangular array of correlations is ambiguous; the authors should define whether it means the limit over pairs with both indices tending to infinity or the limit of the minimum off-diagonal entry of Σ_n.","section":"Section 4"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's own prior work, including an unpublished PhD thesis, for parts of the proofs in Sections 4 and 5. This is not by itself disqualifying, but the present manuscript should be self-contained enough that the main theorems do not depend on assertions that are only sketched by reference. The positive-limit section and the general-correlation extension both need substantial repair; if the author can fix these, the zero-limit contribution for the equicorrelated factor model may be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe main thing to know: the paper's central result is real. For the equicorrelated setup with idiosyncratic density f that is non-increasing on R+ and superexponential (f(x)/f(x-b)→0), Bonferroni FWER goes to zero under positive correlation. The proof conditions on the common factor, does a L'Hopital argument, and works, aside from typos (n→0 in the theorem statements, an α vs d slip in Theorem 3.1). That is a genuine extension of the normal-only results and worth knowing.\n\nThe rest of the paper is a lot shakier. The Pareto example that is supposed to give a strictly positive FWER limit is wrong: the inequality (1 - d/n^{1+δ/2})^n ≥ (1 - d/n^{1+δ/2})^{n^{1+δ/2}} points the wrong way, and the left side actually tends to 1. Worse, the claim can't be fixed while keeping the finite-variance assumption: cBon ~ sqrt(n), and for any finite-variance Z, 1-F(sqrt(n)) = o(1/n), so F^n(sqrt(n)) → 1. The positive-limit idea is not just miscalculated; it's incompatible with the model's unit-variance assumption.\n\nTheorem 4.1, the extension to general positive correlation, also breaks. The proof defines M_n as the set of indices whose correlations with all others are at least δ. The claim that liminf ρ_ij = δ > 0 makes n - |M_n| finite is false. Take ρ_{1j}=1/2 for all j and ρ_{ij}=1 for i,j≥2; then liminf is 1, but M_n is empty for every n. The reduction to the equicorrelated theorem fails. Same issue hits Theorem 5.3: it compares the actual correlation matrix to an equicorrelation matrix at δ, but liminf doesn't give you uniform pairwise correlation at δ. The f/g conditions are also not defined for the general elliptical case, so that transfer is incomplete.\n\nWhat the paper does well is Section 3.1. The zero-limit theorem is a clean, self-contained result. The step-down extension would be substantial if the comparison inequality were justified, but it isn't as written.\n\nThis is a mixed manuscript: a correct core theorem surrounded by overclaims. I'd send it to review — a referee can identify the gaps and the author can either fix them or cut the unsupported parts. For my own work, I wouldn't cite it in its current form, but I'd keep an eye on a revised version.","headline":"The zero-FWER theorem for superexponential equicorrelated statistics is correct; the positive-correlation extension and the positive-limit example are not, as written.","tokens_in":14055,"tokens_out":14484,"would_cite":false,"duration_ms":132688,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62J15","62F03"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that under positive equicorrelation and superexponentially light-tailed margins, the Bonferroni FWER tends to zero as the number of hypotheses grows, and extends this to all step-down procedures and positively…","keywords":["family-wise error rate","multiple testing under dependence","stepwise procedures","Holm's method","elliptically contoured distributions","Bonferroni procedure","equicorrelation","asymptotic FWER"],"falsifier":"Simulate the equicorrelated model with $Z_i$ i.i.d. standard exponential (standardized to mean 0, variance 1) and $U\\sim N(0,1)$, set $\\alpha=0.05$, $\\rho=0.5$, and estimate $\\mathrm{FWER}_{\\mathrm{Bon}}$ by Monte Carlo for $n=10^3,10^4,10^5,10^6$. Since $f(x)/f(x-b)=e^{-b}$ does not tend to 0, condition (ii) fails; if the estimated FWER stays above, say, $10^{-3}$ rather than trending to 0, the paper's claim that a wide class of exponential-family distributions satisfies its conditions is refuted, and the zero-limit theorem is not applicable to Laplace or exponential margins.","tokens_in":12923,"feed_emoji":"📉","tokens_out":9780,"duration_ms":83348,"temperature":0.7,"pith_summary":"The paper asks how positive correlation among test statistics changes the family-wise error rate (FWER) of multiple testing procedures when the test statistics are not assumed Gaussian. It establishes that, for a sequence of equicorrelated test statistics built from independent components with sufficiently light-tailed densities, the Bonferroni procedure's FWER tends to zero as the number of hypotheses grows, under any mix of true and false nulls. The same zero limit is extended to positively correlated elliptically contoured distributions and to every step-down procedure that controls FWER at level α. The paper also constructs distributions, such as Pareto-tailed ones, for which the Bonferroni FWER has a strictly positive limit, showing that the tail condition is the dividing line. If correct, the results imply that in large simultaneous testing problems with positive dependence and light tails, false rejections become asymptotically negligible even with the simplest correction.","feed_headline":"Bonferroni error rate falls to zero under correlation and light tails","feed_subtitle":"New proof covers non-normal test statistics and all step-down procedures, not just the Gaussian case.","key_machinery":"The argument is carried by an explicit upper bound on the Bonferroni FWER (Theorem 3.1): for every $d\\in(0,1)$, $\\mathrm{FWER}_{\\mathrm{Bon}}(n,\\alpha,\\rho) \\le 1 - G(0)F^n(c_{\\mathrm{Bon}}/\\sqrt{1-\\rho}) - [G(c_{\\mathrm{Bon}}(1-d)/\\sqrt{\\rho}) - G(0)]F^n(d\\,c_{\\mathrm{Bon}}/\\sqrt{1-\\rho})$, obtained by conditioning on the common factor $U$ and splitting the integral at 0. Lemma 3.1 then evaluates $\\log\\lim_n F^n(c_{\\mathrm{Bon}}/\\sqrt{1-\\rho})$ as $-\\alpha$ times the limit of a density ratio $f(c_{\\mathrm{Bon}}/\\sqrt{1-\\rho}) / \\int f((c_{\\mathrm{Bon}}-\\sqrt{\\rho}u)/\\sqrt{1-\\rho})g(u)\\,du$. The superexponential tail condition (ii), $f(x)/f(x-b)\\to 0$ for all $b>0$, forces this ratio to zero, so $F^n(\\cdot)\\to 1$ and the upper bound collapses to zero. The same tail condition is the engine behind the elliptical and step-down extensions, via the quadrant-probability comparison of DasGupta et al. and the cut-off optimality of Holm's procedure.","core_discovery":"The central claim is Theorem 3.3 (with Theorem 3.2 for the global null): in the equicorrelated model $X_i = \\sqrt{1-\\rho}Z_i + \\sqrt{\\rho}U$, where $Z_i \\sim F(\\mu_i/\\sqrt{1-\\rho},1)$ are independent and $U\\sim G(0,1)$ is common, if $G(a)<1$ for some $a>0$ and the density $f$ of $F$ is non-increasing on $\\mathbb{R}_+$ with $f(x)/f(x-b)\\to 0$ for every $b>0$, then $\\lim_{n\\to\\infty} \\mathrm{FWER}_{\\mathrm{Bon}}(n,\\alpha,\\rho)=0$ for any $\\alpha\\in(0,1)$, $\\rho\\in(0,1)$, and any configuration of true and false null hypotheses. The proof rests on an upper bound obtained by conditioning on the common factor $U$, and on a lemma showing that under the tail condition the bound's key factor $F^n(c_{\\mathrm{Bon}}/\\sqrt{1-\\rho})$ converges to 1. The same mechanism extends Theorem 4.1 to elliptical densities with $\\liminf \\rho_{ij}>0$, and Theorem 5.3 to all step-down FWER-controlling procedures: the probability of rejecting at least one hypothesis goes to zero. The positive-limit construction (Theorem 3.4) uses a Pareto-type density with variance 1, for which $F^n(\\sqrt{n/(\\gamma(1-\\rho))})$ stays below 1, yielding $\\liminf \\mathrm{FWER}>0$.","pith_inferences":["If the result is right, the practical regime where Bonferroni is 'too conservative' under positive dependence is even broader than the Gaussian case: any light-tailed data with a common factor will have vanishing false-rejection probability as $n$ grows, so researchers can safely use Bonferroni to suppress false positives even at enormous hypothesis counts.","The tail condition (ii) appears to be a sharp phase boundary: it separates distributions for which the proof gives a zero limit from those, like Pareto or exponential, where the limit may be positive. A natural next step is to map the exact critical tail index for intermediate distributions such as Weibull with shape between 0 and 1.","The same conditioning argument could be re-run for two-sided tests or for false-discovery-rate procedures; the zero-limit phenomenon may extend to FDR control, where the analogue would be that the FDP collapses to zero under positive equicorrelation and light tails.","The elliptically contoured extension suggests the result is not an artifact of the additive common-factor structure; any positive dependence that keeps all pairwise correlations bounded below by $\\delta>0$ may inherit the zero limit, so the phenomenon is robust to the precise form of dependence."],"forward_implications":["In the equicorrelated normal case, the known zero-limit results are recovered as special cases, with the new proof covering arbitrary non-Gaussian light-tailed margins.","Any step-down FWER-controlling procedure, including Holm's method, makes no rejection (of true or false nulls) in the limit: both FWER and AnyPwr vanish when $\\liminf \\rho_{ij}>0$ and the tail condition holds.","The upper bound of Theorem 3.1 is asymptotically sharp: it converges to zero exactly when the tail condition holds, and the Pareto example shows the limit can be strictly positive when it fails.","With positive correlation, the Bonferroni threshold becomes increasingly conservative as $n$ grows: the probability of even one false rejection goes to zero, so power, not error control, becomes the binding constraint in large-scale testing."],"supporting_citations":[{"why":"Supplies the Gaussian equicorrelated upper bound $\\alpha(1-\\rho)$ that this paper generalizes and improves.","marker":"Das and Bhandari (2021)"},{"why":"Proves the zero limit for equicorrelated and general correlated normal test statistics, the target of generalization.","marker":"Dey and Bhandari (2023b)"},{"why":"Proves the zero limit for step-down procedures under normal correlation; extended here to elliptical setups.","marker":"Dey (2024b)"},{"why":"Provides the sequence model framework and product-type inequalities for maxima used in the bound.","marker":"Finner and Roters (2001)"},{"why":"Introduces the equicorrelated normal asymptotics for Bonferroni and the setup used throughout.","marker":"Proschan and Shaw (2011)"},{"why":"Gives the quadrant probability comparison for elliptically contoured distributions used in Theorem 4.1.","marker":"DasGupta et al. (1972)"},{"why":"The optimality result that any step-down procedure has $u_1 \\le \\alpha/n$, used in Theorem 5.3.","marker":"Gordon and Salzman (2008)"},{"why":"Defines the step-down procedure whose rejection event bounds all step-down procedures.","marker":"Holm (1979)"}],"fun_headline_variants":["Bonferroni error rate falls to zero for correlated tests","Non-normal test statistics: dependence no longer breaks FWER","Correlated multiple testing: error probability vanishes asymptotically","Step-down procedures also yield zero error under dependence","Light tails and equicorrelation: Bonferroni FWER -> 0"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole zero-limit result hinges on the assumption that the density of the idiosyncratic component has superexponentially light tails: $f$ must be non-increasing on $\\mathbb{R}_+$ and satisfy $f(x)/f(x-b)\\to 0$ for every fixed shift $b>0$. This excludes exponential, gamma, and polynomial-tailed distributions; if it fails, the lemma that drives $F^n(\\cdot)\\to 1$ no longer applies, and the FWER limit is not known to be zero.","fun_headline_variants_meta":{"raw":{"variants":["Bonferroni error rate falls to zero for correlated tests","Non-normal test statistics: dependence no longer breaks FWER","Correlated multiple testing: error probability vanishes asymptotically","Step-down procedures also yield zero error under dependence","Light tails and equicorrelation: Bonferroni FWER -> 0"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00054,"raw_usage":{"total_tokens":2668,"prompt_tokens":1104,"completion_tokens":1564,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":720,"completion_tokens_details":{"reasoning_tokens":1482}},"tokens_in":720,"tokens_out":1564,"duration_ms":13070,"temperature":1.0,"reasoning_tokens":1482,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:55:05.137417+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the equicorrelated model with $Z_i$ i.i.d. standard exponential (standardized to mean 0, variance 1) and $U\\sim N(0,1)$, set $\\alpha=0.05$, $\\rho=0.5$, and estimate $\\mathrm{FWER}_{\\mathrm{Bon}}$ by Monte Carlo for $n=10^3,10^4,10^5,10^6$. Since $f(x)/f(x-b)=e^{-b}$ does not tend to 0, condition (ii) fails; if the estimated FWER stays above, say, $10^{-3}$ rather than trending to 0, the paper's claim that a wide class of exponential-family distributions satisfies its conditions is refuted, and the zero-limit theorem is not applicable to Laplace or exponential margins.","supporting_citations":[],"review_version":1}