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REVIEW 4 major objections 5 minor 1 cited by

Testing Most Influential Sets

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2510.20372 v4 pith:WKIITVDM submitted 2025-10-23 stat.ML cs.LGecon.EMmath.STstat.MEstat.TH

Lucas D. Konrad , Nikolas Kuschnig This is my paper
classification stat.MLcs.LGecon.EMmath.STstat.MEstat.TH MSC 62G3262J0562F03
keywords mostinfluentialsetsinfluencefunctionsextremevaluetheoryFréchetdistributionGumbellinearregressionhypothesistestingsensitivityanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [§3.2.1 (Theorem 1) vs. §3.3 (Implementation)] 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.
  2. [§3.2.2 (Theorem 2)] 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.
  3. [Appendix A3.1 (Lemma 2 and Corollary 2)] 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.
  4. [§3.2.1 proof sketch] 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.
minor comments (5)
  1. [§3.3] 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.
  2. [§3.2.2] 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.
  3. [§4.2.3, §A6] 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.
  4. [Reproducibility statement] The statement says code 'will be made available', but no link or supplement is provided. The empirical results cannot currently be reproduced.
  5. [§3.2.1] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: EVD claims are derived from stated tail assumptions, not fitted; the only self-citation is non-load-bearing.

full rationale

Walking the derivation chain, Proposition 1 is proved recursively from the exact influence formula, and Theorem 1 derives the Fréchet limit from explicit tail assumptions on X and R rather than from any fitted quantity. The inverse-sum and product-EVD lemmas in the Appendix are standard regular-variation/central-limit arguments; they do not define the target Δmax in terms of the fitted parameters. The block-maxima MLE in Section 3.3 estimates location/scale from data blocks excluding S_max^k, which is an empirical null-estimation procedure, not a construction in which the reported p-value is equal to a fitted parameter by definition. The citations to Kuschnig et al. (2021) are motivational/contextual (identifying candidate sets in the economics application) or algorithmic (greedy search), and are not used to establish the extreme-value theorems. There is an internal inconsistency (Theorem 1 states shape ξ = min{ξ_x, ξ_r}, while Section 3.3 says max) and a correctness concern that Theorem 2 applies a CLT to what is effectively an order-statistic sum; however, those are mathematical issues, not circularity. No circular step is exhibited, so the score reflects only the presence of a minor, non-load-bearing self-citation.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities. The framework rests on standard EVT machinery plus domain assumptions about X and R tails; the two ad-hoc assumptions in the theorem proofs are the fragile links in the central derivation.

free parameters (3)
  • k (subset size) = chosen by analyst; fixed or k=pN
    The paper's two regimes depend on how k scales with N; k is not estimated but is a user choice.
  • M (number of blocks) = chosen by analyst
    Block maxima MLE uses an arbitrary M; the Gumbel bias correction depends on M.
  • Tail coefficient estimation threshold = not specified
    Shape parameter ξ is estimated via MLE (Smith 1985); the threshold for 'sufficiently close to zero' in §3.3 is arbitrary.
assumptions (5)
  • domain assumption Features X and residuals R are independent.
    Acknowledged in §5.1: 'Our analysis leverages independence between features and the residuals, which can be restrictive in practice.' It underpins the product EVD derivations.
  • domain assumption Xi and Ri have polynomial tails with coefficients ξx and ξr.
    Assumed in Theorem 1; needed for the Fréchet regime.
  • ad hoc to paper The numerator of the maximizing set behaves like the tail of max{XiRi}.
    In the proof sketch of Theorem 1: 'its upper tail behaves like the tail of max{XiRi} for i∈S^max_k'. This tail-equivalence for the subset-maximizer is asserted, not proven.
  • ad hoc to paper C = Σ XiRi and D_{-S}^{-1} are asymptotically independent.
    Stated in the proof sketch of Theorem 1 without a formal justification; the residual/feature dependence may break it.
  • ad hoc to paper For |S|=pN, the CLT applies to the numerator of the maximizing subset.
    Theorem 2 proof: 'By the CLT, C/√N ∼ N(μ,σ²)'. The maximizing subset is not a random subset; it is the set of largest products, so the CLT does not apply.

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Cite this review

Pith. "Pith review of Testing Most Influential Sets." pith.science (2026). https://pith.science/paper/WKIITVDM

@misc{pith2026251020372,
  author       = {Pith},
  title        = {Pith review of: Testing Most Influential Sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WKIITVDM}},
  note         = {Machine review of arXiv:2510.20372}
}
read the original abstract

Small influential data subsets can dramatically impact model conclusions, with a few data points overturning key findings. While recent work identifies these most influential sets, there is no formal way to tell when maximum influence is excessive rather than expected under natural random sampling variation. We address this gap by developing a principled framework for most influential sets. Focusing on linear least-squares, we derive a convenient exact influence formula and identify the extreme value distributions of maximal influence - the heavy-tailed Fr\'echet for constant-size sets and heavy-tailed data, and the well-behaved Gumbel for growing sets or light tails. This allows us to conduct rigorous hypothesis tests for excessive influence. We demonstrate through applications across economics, biology, and machine learning benchmarks, resolving contested findings and replacing ad-hoc heuristics with rigorous inference.

Figures

Figures reproduced from arXiv: 2510.20372 by the authors.

Figure 1
Figure 1. Illustration of our methodology on a simple linear regression with a moderately influential [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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Reference graph

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