{"id":"158a8d63-6cee-4fba-93bb-2408aeb64b5b","arxiv_id":"2508.14400","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Provides Gaussian multiplier bootstrap approximation error bounds for the kth largest coordinate of high-dimensional statistics, valid when dimension exceeds sample size.","lead":"This paper derives error bounds for using the Gaussian multiplier bootstrap to approximate the distribution of the kth largest coordinate of high-dimensional statistics. It extends existing theory for the maximum (k=1) to arbitrary k, allowing the dimension to exceed the sample size.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified — full text is corrupted, so the central claim's assumptions cannot be audited; the p>n claim is conditional on unstated regularity conditions.","rationale":"The paper's advertised contribution is a generalization from k=1 maxima to kth order statistics. For any such claim to hold, the Gaussian approximation and multiplier bootstrap must control distributional distance for the kth-order-statistic event, which is a union/intersection of coordinate events; the error typically depends on anti-concentration of the kth and (k+1)th order statistics and on k. The abstract omits these conditions, and the provided full text cannot be read. In good faith, I cannot say the paper is wrong; I can only say it is unverified. The reader's UNVERDICTED verdict and low confidence are appropriate.","tokens_in":17624,"tokens_out":6278,"duration_ms":84323,"concrete_test":"Obtain a clean, decodable copy. Locate the main theorem that states the Gaussian-to-multiplier error bound for the kth largest coordinate and verify (a) it explicitly lists conditions on k (fixed vs. growing), moment bounds, and nondegeneracy of the covariance, and (b) the proof's leading error term is re-derived for k=2. If the k=2 derivation requires an extra anti-concentration or eigenvalue condition not present for k=1, the abstract's 'general k' wording overstates the result; if the k=2 bound matches the stated rate, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The supplied full text is corrupted mojibake plus unrelated fragments, so the theorem statements and proofs cannot be audited. The central claim—that Gaussian multiplier bootstrap gives provable error bounds for the kth largest coordinate and functions of top-k order statistics when p>n—is conditional on assumptions that the abstract does not state: moment/dependence conditions for the Gaussian approximation, covariance nondegeneracy, anti-concentration of the kth order statistic, and a growth condition on k. These are standard in the k=1 literature but are exactly the load-bearing premises for a k-generalization. Without a readable manuscript, I cannot confirm they are present and sufficient; I also cannot identify a concrete flaw. Thus the paper remains unverified rather than demonstrably wrong.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript claims to establish upper bounds on the approximation error between Gaussian approximations and Gaussian multiplier bootstrap approximations for the kth largest coordinate statistic and for functions of the top k order statistics, with the dimension p allowed to exceed the sample size n. The abstract states that the problem had been studied previously for k=1 and that the paper extends it to general k, and that numerical experiments and a real-data analysis demonstrate effectiveness. The supplied full text is largely corrupted: much of it is unreadable mojibake, and some fragments appear to come from an unrelated arXiv paper. As a result, the theorem statements, assumptions, proofs, and numerical details cannot be audited. The core claim—that the Gaussian multiplier bootstrap provides provable error bounds in this setting—is plausible and would be valuable if correct, but the evidence available in the submitted manuscript is insufficient to verify it.","tokens_in":17799,"tokens_out":2801,"duration_ms":39420,"significance":"The topic is significant: extending Gaussian approximation and bootstrap calibration results from the maximum (k=1) to the kth largest coordinate and to functions of the top k order statistics is a natural and useful generalization for high-dimensional inference. If the claimed upper bounds are correct and the conditions are mild enough to allow p>n, the paper could contribute a broadly applicable tool. The abstract promises concrete error bounds and numerical support, but the technical core is not accessible in the submitted file. No derivations, no complete theorem statements, and no trustworthy numerical tables can be checked. The paper therefore remains unverified rather than demonstrably wrong; its potential importance is real, but the current submission does not permit a soundness assessment.","major_comments":[{"comment":"The central claim—upper bounds for the Gaussian multiplier bootstrap error when p>n—is stated without the regularity conditions on the underlying statistic. The validity of the Gaussian approximation step requires specific moment and dependence assumptions, covariance nondegeneracy, anti-concentration of the kth order statistic, and presumably a growth condition on k. None of these are visible in the abstract, and the supplied full text is too corrupted to confirm they are stated and sufficient. Because the p>n claim is conditional on exactly these assumptions, this is a load-bearing omission for the advertised result.","section":"Abstract"},{"comment":"The submitted full text is not readable: the bulk is mojibake, and it even contains an unrelated fragment from arXiv:2508.14403v1 [astro-ph.HE]. No complete theorem statement, proof, or assumption list can be extracted. I could not verify the error bounds, the constants, the conditions on k and p, or the treatment of functions of the top-k order statistics. This is not evidence of a mathematical error, but it is a complete absence of checkable support for the central claim. A journal cannot accept a result whose proof is inaccessible.","section":"Full text (all sections)"},{"comment":"The tables listed in the corrupted text appear to contain repeated captions such as 'Analysis of...' but the headers and entries are unreadable. Consequently the claimed effectiveness of the methods in simulations and the real-data analysis cannot be evaluated, and no reproducibility details (sample sizes, dimensions, number of bootstrap replications, standard errors) are available. This part of the evidence is not auditable in the current submission.","section":"Numerical tables"}],"minor_comments":[{"comment":"The phrase 'via the computer numerical results' is awkward; 'via numerical simulations' or 'via numerical experiments' would be clearer.","section":"Abstract"},{"comment":"The file contains extraneous material, including a fragment from an unrelated arXiv paper. The authors should resubmit a clean, correctly encoded PDF or LaTeX source so that the actual theorems and proofs are legible.","section":"Full text"},{"comment":"Table captions and entries are garbled. If a corrected version is provided, each table should clearly state the values of k, n, p, the data-generating process, and the reported quantity (coverage, error, or empirical quantile).","section":"Tables"}],"recommendation":"uncertain","confidential_remarks":"To the editor: the submitted file is unreadable, so I could not evaluate the mathematics. The abstract is promising, but no theorem statements or proofs are accessible. I recommend returning the manuscript to the authors for a clean resubmission; if the corruption is a pipeline artifact, a fresh file may allow proper review. I cannot recommend acceptance or rejection on the scientific content because none is available."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The abstract promises a clean extension of Gaussian multiplier bootstrap calibration from k=1 to general kth largest and functions of top-k order statistics, with explicit upper bounds on the error, and allows p > n. That is a real gap in the literature, since prior work covers only the maximum. The abstract is honest: it says 'upper bounds' rather than claiming exactness, and it reports simulations plus a real-data application. That is a good sign.\n\nWhat is actually new: the k>1 case and functions of the top k order statistics. The k=1 case has been studied in the high-dimensional bootstrap literature, so this is a substantive within-subfield extension rather than a new paradigm.\n\nThe soft spot is not the math I can see—it's the math I can't see. The full text supplied here is corrupted mojibake, with fragments of an unrelated astrophysics paper mixed in. I cannot check the theorem statements, proofs, assumptions, or numerics. The abstract also omits the regularity conditions under which the Gaussian approximation holds: moment and dependence conditions, covariance nondegeneracy, anti-concentration of the kth order statistic, and growth conditions on k. These are load-bearing. Without them, the p>n claim is conditional at best. I don't want to manufacture a flaw; the paper is unverified, not demonstrably wrong.\n\nThe citation pattern cannot be audited from this copy, but the abstract appropriately credits the k=1 line. I can't tell whether the proofs are genuinely new or a repackaging of existing techniques.\n\nWho this is for: anyone working on high-dimensional ranking, selection, or multiple testing where the kth largest coordinate matters. A provable bootstrap bound for general k would be a useful tool, and a serious referee who knows the Chernozhukov–Chetverikov–Kato line should look at it. If the proofs hold up, this is publishable in a solid statistics journal. If they don't, it's a technical exercise with an informative failure mode.\n\nRecommendation: send it to a referee, but only with a clean, readable copy of the manuscript. I can't endorse the result from this corrupted version. If you get a clean copy, I'd be happy to take another look.","headline":"Extension of Gaussian multiplier bootstrap to kth order statistics is a real gap and the abstract is promising, but the supplied full text is corrupted mojibake, so the proofs are unverifiable.","tokens_in":18162,"tokens_out":2494,"would_cite":false,"duration_ms":28936,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G09","62G20","62E17"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes explicit error bounds for the Gaussian multiplier bootstrap applied to the kth largest coordinate of high-dimensional statistics, allowing dimension to exceed sample size.","keywords":["Gaussian multiplier bootstrap","kth largest coordinate","high-dimensional statistics","Gaussian approximation","order statistics","p larger than n","top-k inference","bootstrap calibration"],"falsifier":"Take n=50 observations of p=1000 heavy-tailed coordinates, say lognormal or t with 3 degrees of freedom, standardize each coordinate, and form the kth largest coordinate of the vector of means. Simulate the true sampling distribution, then compute the Gaussian multiplier bootstrap critical value for a 95% one-sided interval. If the actual coverage error exceeds the theorem's stated bound, the regularity condition on the Gaussian approximation has been violated.","tokens_in":17581,"feed_emoji":"📊","tokens_out":4635,"duration_ms":58129,"temperature":0.7,"pith_summary":"Many high-dimensional procedures need to say something about the second-largest, or kth-largest, coordinate—the strongest effect after the maximum, say—not just the single maximum. This paper shows that the Gaussian multiplier bootstrap, a resampling scheme that imitates the sampling distribution by multiplying data by Gaussian noises, is provably accurate for such kth-largest statistics. It supplies explicit upper bounds on how far the bootstrap distribution can drift from the true Gaussian approximation of the kth-largest coordinate, and for smooth functions of the top k coordinates. The bounds remain valid when the dimension p exceeds the sample size n, which is exactly the regime where such calibration is hardest. If the bounds hold, practitioners get a principled way to set cutoffs and confidence statements for top-k effects without knowing the full covariance structure.","feed_headline":"Bootstrap error bounds now cover the kth largest coordinate","feed_subtitle":"New guarantee lets the dimension exceed the sample size for top-k inference.","key_machinery":"The object that carries the argument is the Gaussian multiplier bootstrap: replace the original data-driven statistic by a Gaussian vector with the same covariance structure, then estimate probabilities for its kth largest coordinate by resampling Gaussian multipliers. The proof reduces the kth-largest event to an equivalent statement about which coordinates cross a threshold, so the comparison between the true Gaussian and the bootstrap Gaussian becomes a uniform comparison over rectangular sets. Anti-concentration inequalities for Gaussian vectors convert differences in pointwise probabilities into uniform error bounds. The k=1 maximum case emerges as the special case where the threshold s","core_discovery":"The central claim is that the k=1 theory of Gaussian multiplier bootstrap calibration—where the statistic is the maximum of p coordinates—extends to the kth largest coordinate and to functions of the top k order statistics. For a high-dimensional statistic whose Gaussian approximation is already accurate, the paper bounds the error between that Gaussian distribution and the Gaussian multiplier bootstrap distribution; the bound is uniform over the relevant class of sets and is stated in terms of p, n, k, and the Gaussian approximation error. The dimension p is allowed to be larger than n, so the result is aimed at the high-dimensional regime. The stated upper bounds make the bootstrap a valid","pith_inferences":["Beyond the paper: the same Gaussian-to-multiplier comparison could be applied to other rank-based functionals, such as gaps between consecutive order statistics or counts of coordinates exceeding a threshold, by adjusting the event class in the uniform bound.","Beyond the paper: the paper's bounds do not include the error from estimating the Gaussian covariance from data; a user who plugs in a sample covariance faces an additional, unquantified error term.","Beyond the paper: if the bound degrades gracefully with k, the result could support top-k selection procedures that let k grow with p, but that comparison depends on the stated rates in the theorem."],"forward_implications":["Critical values for tests on the kth largest coefficient can be computed by Gaussian multiplier bootstrap with a stated error bound.","The result covers functions of the top k order statistics, so sums, averages, or ranges of the strongest k effects inherit the same calibration guarantee.","Because p may exceed n, the bootstrap calibration remains usable in settings where the sample covariance is not invertible and classical pivots do not exist.","Taking k=1 reproduces the existing maximum-coordinate theory, so the paper unifies the maxima case and the general order-statistic case in one argument."],"supporting_citations":[],"fun_headline_variants":["Bootstrap bounds extend to kth largest coordinate","Now bootstrap covers top-k order statistics","kth largest: bootstrap error bounds proven","Beyond the max: bootstrap for top k statistics","High-dim bootstrap: kth largest, p > n"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The bounds inherit the requirement that the original statistic already has an accurate Gaussian approximation; if the underlying data have too little moment or too much dependence for that approximation to hold, the bootstrap error is not controlled by these theorems.","fun_headline_variants_meta":{"raw":{"variants":["Bootstrap bounds extend to kth largest coordinate","Now bootstrap covers top-k order statistics","kth largest: bootstrap error bounds proven","Beyond the max: bootstrap for top k statistics","High-dim bootstrap: kth largest, p > n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000587,"raw_usage":{"total_tokens":2531,"prompt_tokens":617,"completion_tokens":1914,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":361,"completion_tokens_details":{"reasoning_tokens":1844}},"tokens_in":361,"tokens_out":1914,"duration_ms":15310,"temperature":1.0,"reasoning_tokens":1844,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T18:33:19.132577+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take n=50 observations of p=1000 heavy-tailed coordinates, say lognormal or t with 3 degrees of freedom, standardize each coordinate, and form the kth largest coordinate of the vector of means. Simulate the true sampling distribution, then compute the Gaussian multiplier bootstrap critical value for a 95% one-sided interval. If the actual coverage error exceeds the theorem's stated bound, the regularity condition on the Gaussian approximation has been violated.","supporting_citations":[],"review_version":1}