{"id":"81ddf384-067b-4a8c-99b4-e3cccb7f4ea2","arxiv_id":"2510.20372","paper_version":4,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"Maximum influence of the most influential k-point subset in OLS follows a Fréchet distribution (heavy tails, fixed k) or Gumbel distribution (light tails or growing k), enabling tests of excessive influence.","lead":"This paper derives extreme-value distributions for the maximum influence of small data subsets in linear regression, claiming fixed-size sets with heavy tails follow a Fréchet law and growing sets or light tails follow a Gumbel law. It uses these laws to test whether an influential set's impact is excessive, with applications to economics, biology, and ML benchmarks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's Gumbel claim for growing sets is unsupported: the maximizing subset is an order-statistic sum, so Δmax is asymptotically Normal around a constant, not Gumbel.","rationale":"The paper's central contribution is the pair of EVD theorems that justify the hypothesis tests. The reader's weakest assumption pinpoints Theorem 2, and my reading agrees: the asymptotic argument in Section 3.2.2 misidentifies the object. The numerator of Δmax for a growing set is not a sum over a random subset; it is the sum of the top pN order statistics of Z_i=X_iR_i. For finite-variance Z, the Bahadur–Kiefer/empirical-process representation gives √N fluctuations around a deterministic constant, so the centered and scaled Δmax converges to a Normal, not a Gumbel. Corollary 2 does not rescue this because its hypotheses require S to be in MDA(Λ), but the order-statistic sum is in the Gaussian domain, not the Gumbel maximum domain. The consequence is not a minor technical gap: the block-maxima/Gumbel calibration in Sections 3.3 and 4.1.2 is used to obtain p-values for the growing-set regime (e.g., Adult Income). If the true limiting behavior is Normal on the √N scale, those p-values are not valid. I also note the internal contradiction between Theorem 1's min and Section 3.3's max, which independently undermines the constant-size testing recipe as written. Because the central testing framework rests on these theorems, the REJECT verdict is appropriate; no adjustment is needed. The proposed Monte Carlo is a direct falsification test of Theorem 2's Gumbel claim and would settle whether this concern lands.","tokens_in":15272,"tokens_out":10129,"duration_ms":88573,"concrete_test":"Monte Carlo check: set N=5000, p=0.1, draw X_i,R_i iid N(0,1), compute Δmax using the set of the pN largest X_iR_i (for a small N, e.g., N=30, first verify by exhaustive enumeration that this set equals the exact maximizer of Δ(S)). Repeat ≥2000 times. Compute T_N = √N (Δmax - mean)/sd and compare its empirical distribution with N(0,1) and with a fitted Gumbel (QQ plot or Anderson–Darling). If T_N is Normal rather than Gumbel, Theorem 2's Gumbel claim is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim for relative-size sets (Theorem 2, Section 3.2.2) is not supported and appears false. For k=pN, Smax selects the largest pN products Z_i=X_iR_i (up to second-order denominator effects), not a random sample. The proof sketch says 'By the CLT, C/√N ∼ N(μ,σ²)' and then invokes Corollary 2 to reach a Gumbel limit. But that CLT, correctly applied to the order-statistic sum, gives C = N a + √N G_N + o_p(√N) with G_N asymptotically Normal, so Δmax = C/D_{-S} = a/[(1-p)E[X²]] + N^{-1/2} * Normal + o_p(N^{-1/2}). The limit distribution on the √N scale is Normal, not Gumbel; with fixed location/scale the raw Δmax is degenerate at a constant. The Appendix does not prove Theorem 2; Lemmas 2 and Corollary 2 concern products of a light-tailed variable with a regularly varying variable, not maxima over dependent subset sums. This is load-bearing: the Adult Income application and the claim that growing sets are 'well-behaved Gumbel' feed directly into the reported p-values. (Separately, Theorem 1 states shape ξ=min{ξ_x,ξ_r} while Section 3.3 says max, an internal contradiction that also needs correction.)","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an extreme-value framework for testing whether a most influential subset in linear regression is excessively influential. It derives an exact closed-form expression for the change in a regression coefficient when a subset S is removed (Proposition 1), and then claims two asymptotic regimes: constant-size sets lead to a Fréchet limit (Theorem 1), while sets growing as pN lead to a Gumbel limit (Theorem 2). The proposed testing procedure estimates tail coefficients, uses block maxima to estimate location and scale, and computes p-values for observed maximum influence. The paper applies this procedure to economic, biological, and machine-learning examples.","tokens_in":15657,"tokens_out":9545,"duration_ms":78651,"significance":"The problem is timely and practically motivated, and Proposition 1's exact subset-influence formula is a useful contribution. If the distributional claims were correct, the paper would provide a principled inferential alternative to ad-hoc sensitivity checks. However, the central results are not supported: Theorem 2's Gumbel claim for growing sets appears false, Theorem 1's shape parameter is internally inconsistent with the implementation, and the appendix's supporting lemmas contain role-reversal errors. The exact formula and simulation setup are assets, but they do not compensate for the invalid theory on which the hypothesis tests rest.","major_comments":[{"comment":"Theorem 1 states that for constant-size sets the limiting Fréchet shape is ξ = min{ξ_x, ξ_r}, while Section 3.3 implements ξ = max{ξ_x, ξ_r} and uses the quantity 1/max{ξ_x, ξ_r}. The paper never defines ξ_x and ξ_r explicitly. Under the standard convention that ξ is a Fréchet shape (tail index α = 1/ξ), the product of independent regularly varying factors with shapes ξ_x, ξ_r has shape max{ξ_x, ξ_r}, not min. Under the alternative convention that ξ is a tail exponent, the min is the correct exponent but the Fréchet shape is 1/min, not min. Either way, Theorem 1 and Section 3.3 cannot both be correct. Since the test's distributional family and p-values depend on this choice, this is a load-bearing inconsistency.","section":"§3.2.1 (Theorem 1) vs. §3.3 (Implementation)"},{"comment":"The proof sketch applies a CLT to C, the numerator of the maximizing subset, writing 'By the CLT, C/√N ∼ N(μ,σ²)'. But C is the sum of the largest pN products X_iR_i, not a random sample. For fixed p∈(0,1) and finite-variance products, the sum of the top pN order statistics satisfies C = N a_p + √N G_N + o_p(√N), where G_N is asymptotically Normal. Consequently Δmax = C/D_{-S} is degenerate at a_p/((1-p)E[X²]) with √N-scale Normal fluctuations; it is not Gumbel. The appendix contains no proof of Theorem 2; Lemmas 2 and Corollary 2 concern independent products, not maxima over dependent subset sums. This invalidates the Gumbel claim used for the Adult Income application and for all relative-size-set testing.","section":"§3.2.2 (Theorem 2)"},{"comment":"Lemma 2 is stated as: if T∈MDA(Λ) and S∈MDA(Φ_a), then Δ=S·T∈MDA(Φ_a). The proof, however, says 'for Gumbel tails (S)' and 'for the Fréchet tails (T)', swapping the roles of S and T. The derived regularly varying tail comes from the factor that is actually Fréchet in the proof, not from the factor that the lemma declares Gumbel. Corollary 2 then claims that 'assuming a tail coefficient a=∞ it follows that S∼Gumbel' and concludes Δ∈MDA(Λ), but this does not follow from Lemma 2, whose proof assumes a finite a. The lemma and corollaries therefore do not provide the stated support for Theorems 1 and 2.","section":"Appendix A3.1 (Lemma 2 and Corollary 2)"},{"comment":"The proof of Theorem 1 asserts that C and D^{-1}_{-S} are asymptotically independent and that the upper tail of C behaves like the tail of max{X_iR_i} over S_max. Neither assertion is justified. The denominator D_{-S} depends on S through the removed leverage terms, and S_max is defined by maximizing the ratio C/D_{-S}, not C alone. In addition, residuals R are functionally dependent on X in finite samples; the paper only notes the independence assumption as a limitation in §5.1, without proving the asymptotic independence needed for the theorem. This coupling is central to the claimed Fréchet limit and is not addressed.","section":"§3.2.1 proof sketch"}],"minor_comments":[{"comment":"The threshold for '1/max{ξ_x, ξ_r} sufficiently close to zero' is not defined, and the number of blocks M is a free parameter. The procedure would benefit from concrete guidance or a sensitivity analysis over these choices.","section":"§3.3"},{"comment":"The displayed CLT statement 'C/√N ∼ N(μ,σ²)' is dimensionally inconsistent for an uncentered sum of N terms; the centered version should be (C−Nμ)/√N.","section":"§3.2.2"},{"comment":"Typos and formatting issues: 'Frechét' should be 'Fréchet'; 'We investigate 2and2 data points' is garbled; 'p= 0.014.(See Table A2' has misplaced punctuation; Table A2's 'Full Set4' is unclear.","section":"§4.2.3, §A6"},{"comment":"The statement says code 'will be made available', but no link or supplement is provided. The empirical results cannot currently be reproduced.","section":"Reproducibility statement"},{"comment":"The definitions of ξ_x and ξ_r should appear in one place and be used consistently throughout; the current text leaves the reader to infer whether these are tail exponents or Fréchet shape parameters, which is the direct source of the min/max contradiction.","section":"§3.2.1"}],"recommendation":"reject","confidential_remarks":"The paper has a solid exact-influence formula and an interesting problem, but the theoretical core is not reliable: Theorem 2 is false as stated, Theorem 1 is internally inconsistent with the implementation, and the appendix lemmas contain role-reversal errors. These are not presentation issues; they concern the distributions that define the p-values. I recommend rejection in the current form. A resubmission would need a corrected product-tail derivation (with unambiguous tail indices), a valid treatment of relative-size sets (which will likely be asymptotic normality rather than Gumbel), and a proof that accounts for the data-dependent selection of S_max and the dependence between residuals and design."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe key takeaway: this paper has one clean, correct result — the exact set-influence formula for OLS, Δ(S) = Σ_{i∈S} x_i r_i / Σ_{n∉S} x_n² — and a genuinely useful idea: formal tests for 'excessive' influence via extreme value theory. But the two main theorems that the testing procedure relies on are not sound as stated, and one is likely false. I would not trust the reported p-values without substantial revision.\n\nWhat is actually new: the closed-form formula is a neat simplification of Cook's distance for subsets, and the block-maxima calibration is a sensible empirical idea. The simulations for constant-size sets are encouraging, and the Fréchet behavior in that regime is plausible.\n\nSoft spots, in order of severity:\n\n1. Theorem 2 (growing sets, k=pN) is unsupported. The maximizer is the sum of the top pN products X_i R_i, not a random subset. That sum has the form aN + √N·Normal + o_p(√N), so Δmax is degenerate at a constant on the raw scale; the √N-scaled limit is Normal, not Gumbel. The proof's invocation of the CLT and then 'Lemma 2 and Corollary 2' does not bridge that gap. This is load-bearing: the Adult Income application and any growing-set case rely on it.\n\n2. Theorem 1 and the implementation contradict each other. The theorem states ξ = min{ξ_x, ξ_r}; Section 3.3 uses ξ = max{ξ_x, ξ_r}. That changes the fitted distribution and the resulting p-values. The Appendix Lemma 2 also swaps S and T between statement and proof, which makes checking the argument harder.\n\n3. The block-maxima procedure is a reasonable calibration method, but the paper sells it as theorem-backed. The null distribution is estimated from data that already exclude the most influential set, so the p-values are optimistic. That deserves a clearer caveat.\n\nI agree with the reader's verdict: reject. The exact formula is worth keeping, and the applications are suggestive, but the advertised 'rigorous tests' are not built on sound foundations. A serious referee would be justified in asking for major revision — proof fixes or, at minimum, an honest reframing as empirical calibration. I would still send it to review, because the problem matters and the exact influence formula is a real step forward, but I would not cite it in its current form.\n\nI'd bring it to a reading group as a cautionary example of EVT applied to dependent extremal sums.\n\nBest.","headline":"The exact influence formula is a clean, correct contribution, but the two central EVT theorems — especially the Gumbel claim for growing sets — are not supported, and the paper contradicts its own shape parameter.","tokens_in":16091,"tokens_out":5011,"would_cite":false,"duration_ms":45863,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G32","62J05","62F03"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives a significance test for the most influential data subsets in linear regression, showing that maximal influence follows a Fréchet distribution for fixed-size sets and a Gumbel distribution for growing sets.","keywords":["most influential sets","influence functions","extreme value theory","Fréchet distribution","Gumbel distribution","linear regression","hypothesis testing","sensitivity analysis"],"falsifier":"Simulate heavy-tailed X and R (e.g., t(5) residuals) with k = pN, compute Δmax for many datasets, and fit a generalized extreme-value distribution to block maxima. If the fitted shape parameter does not converge to zero as N grows, Theorem 2's Gumbel claim is falsified; more directly, examine whether the scaled numerator √N(∑_{top pN} X_iR_i − constant) converges to a Normal or to a different law.","tokens_in":15136,"feed_emoji":"📊","tokens_out":3848,"duration_ms":30195,"temperature":0.7,"pith_summary":"The paper tries to settle when the influence of a small set of data points on a regression coefficient is genuine excess rather than ordinary sampling variation. It derives an exact closed form for the influence of any subset, then shows that the influence of the most influential set obeys one of two extreme-value laws: heavy-tailed Fréchet when the set size is fixed and the data have polynomial tails, and light-tailed Gumbel when the set grows proportionally with the sample or tails are light. On that basis it builds a hypothesis test with an extreme-value null distribution, with block-maximum parameter estimation. The authors apply the test to contested economic and biological findings and ML fairness benchmarks, reporting p-values for excessive influence.","feed_headline":"A formal test says when data points are too influential","feed_subtitle":"Linear-regression study derives exact influence formula and p-values for most influential subsets.","key_machinery":"The exact influence formula Δ(S) = (∑_{i∈S} x_i r_i) / (∑_{n∉S} x_n²) — an additive numerator of products and a denominator that removes the subset's own squared features. This formula makes influence of sets as tractable as single-point influence. The distributional results then follow from extreme value theory: heavy-tailed products in the numerator give Fréchet; CLT-regular sums with light tails give Gumbel; the inverse sum of squares converges to Normal by the delta method.","core_discovery":"Central claim: Δmax, the maximum influence of any subset of size at most k, has a limiting extreme value distribution. For constant k, Δmax converges to a Fréchet distribution with shape ξ = min{ξ_x, ξ_r}, where ξ_x and ξ_r are tail coefficients of predictor and residual. For k = pN, Δmax converges to a Gumbel regardless of underlying distributions when the variance of X_i R_i is finite. This enables a formal test of the null hypothesis that observed influence is natural sampling variation, with block-maxima estimation of location and scale parameters.","pith_inferences":["A natural extension is to influence on predictions (X β̂) and to other linear estimators; the paper notes prediction follows trivially, but the extreme-value argument likely extends with the same dichotomy.","Because the test's null distribution is derived from the same data that produces the candidate set, the procedure is best viewed as a calibration tool for 'surprisingness' rather than a controlled test; the block-maxima estimation excluding the candidate set partially addresses this.","The Gumbel claim for growing sets hinges on the numerator being a random-sample sum; if the subset is the largest order statistics of products, the sum's fluctuations have a different scaling, so a simulation check for heavy-tailed products with growing k would clarify whether Gumbel actually holds.","Cross-validation of the test: apply to datasets with known injected outliers and measure false-positive rates across N and k."],"forward_implications":["A practitioner can compute a p-value for the observed influence of any most influential set instead of relying on ad-hoc thresholds.","For fixed-size sets and heavy-tailed data, the Fréchet shape means even enormous influence values are not necessarily unusual; tests need to account for heavy tails.","For growing sets or light-tailed data, the Gumbel null gives a threshold that grows slowly with sample size, making the test more sensitive to gross outliers.","The framework formalizes and corrects the common √N rule-of-thumb for influential coefficients, which the paper shows is asymptotically accurate only for randomly selected subsets, not for maximized ones.","In the case studies, several contested findings (Seychelles effect, sparrow head-bill, Boston crime) are flagged as excessively influenced, while the Law School large set and Adult Income top 1% are not."],"fun_headline_variants":["New test flags when data points sway results too much","When are influential data points just noise? Now we can tell","Formal test reveals if extreme influence is excessive or expected","Fréchet and Gumbel: a statistical test for overly influential sets","Exact influence formula plus p-values for most influential subsets"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Theorem 2 assumes that the numerator of the maximizing subset behaves like a sum of a random sample (so the CLT applies), but the subset is selected by maximizing influence, so its sum is an order-statistic sum whose fluctuations may have different scaling; if that assumption fails, the Gumbel limit for growing sets would need replacement.","fun_headline_variants_meta":{"raw":{"variants":["New test flags when data points sway results too much","When are influential data points just noise? Now we can tell","Formal test reveals if extreme influence is excessive or expected","Fréchet and Gumbel: a statistical test for overly influential sets","Exact influence formula plus p-values for most influential subsets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":8.4e-05,"raw_usage":{"total_tokens":737,"prompt_tokens":622,"completion_tokens":115,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":366,"completion_tokens_details":{"reasoning_tokens":27}},"tokens_in":366,"tokens_out":115,"duration_ms":2209,"temperature":1.0,"reasoning_tokens":27,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T08:26:47.504694+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate heavy-tailed X and R (e.g., t(5) residuals) with k = pN, compute Δmax for many datasets, and fit a generalized extreme-value distribution to block maxima. If the fitted shape parameter does not converge to zero as N grows, Theorem 2's Gumbel claim is falsified; more directly, examine whether the scaled numerator √N(∑_{top pN} X_iR_i − constant) converges to a Normal or to a different law.","supporting_citations":[],"review_version":1}