{"id":"8227e4f0-be4f-45a0-9d9f-e2dfab94a2f1","arxiv_id":"2507.22967","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Local influence diagnostics for extreme-value Birnbaum-Saunders regression identify a catastrophic wind gust as the dominant observation in an Itajaí dataset.","lead":"A statistical tool is introduced for finding which individual weather records most distort an extreme-value model of wind gusts. Applied to Brazilian data, it flags a documented 2017 storm that killed three people as the most influential observation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim inherits an unproved MLE-regularity premise: Section 3.1 sets aside the support-dependent GEV/log-EVBS problem, Section 5.2 admits gamma in (-1,1/4), and the known non-regular regime gamma in (-1,-1/2] is not covered by the Section 4 simulations.","rationale":"The authors' stated contribution is a local-influence toolkit whose 'rigorous identification' rests on standard likelihood asymptotics. In good faith, the derivations of the Delta matrices and Hessian blocks are consistent with Cook and Poon-Poon, and the Itajai case study is a genuinely nice calibration: the largest eigenvalue flags a documented storm, and the leave-one-out change in gamma is large. The paper is not internally inconsistent; it explicitly discloses the regularity assumption in Section 3.1. But disclosure does not remove load-bearing weight. If the MLE is not asymptotically normal over the admitted parameter range, the inverse observed Hessian in Eqs. (21)-(24) does not have its nominal interpretation, and the q/sqrt(n) benchmark in Section 5.1 is not a rigorous threshold. I agree with the reader's weakest assumption. I would not escalate to rejection: the application's gamma = -0.155 is in the regular range, and the simulation results at gamma = -0.2, 0, and 0.2 are consistent with the maintained assumptions. The appropriate fix is a conditional acceptance asking for a proof or a restriction of the parameter space, plus a simulation that exercises the diagnostic itself. The latter matters because Section 4 validates MLE bias, RMSE, and coverage, not whether CNC actually flags planted influential observations; a single post-hoc real-data example cannot by itself establish sensitivity and specificity. That said, the regularity concern is the most fundamental.","tokens_in":28048,"tokens_out":12129,"duration_ms":142222,"concrete_test":"Use the Section 4 Monte Carlo design with gamma = -0.7 and gamma = -0.9, plus gamma = -0.2 as a control, with n = 180 and 5,000 replicates, and compute the empirical coverage of nominal 95% Wald intervals for gamma based on the observed Hessian. If coverage in the gamma = -0.7 and -0.9 cases is substantially below 0.95 while the gamma = -0.2 case is near nominal, then the inverse-Hessian-based CNC thresholds are not valid in that regime, and the paper should either restrict the method to gamma > -1/2 or supply a proof of regularity for the log-EVBS regression model. As a secondary check, compare the application's Wald SE for gamma (0.0472) with a parametric bootstrap SE; a large discrepancy would indicate that even the fitted point is not in a regime where the observed information is trustworthy.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.1 explicitly states that the support of the GEV/log-EVBS distribution depends on the parameters, that verifying standard MLE regularity conditions 'is a problem that deserves to be investigated, but it is far from the objective of this work,' and that the authors simply maintain assumptions used for the GEV family. This is the load-bearing premise for the diagnostic machinery: the normal-curvature formulas in Section 5.1 use the inverse observed Hessian, and the Poon-Poon reference threshold q/sqrt(n) is only meaningful if the MLE is consistent and asymptotically normal with covariance estimated by the inverse observed information. The paper's own working assumptions are gamma > -1 (for consistency) and gamma < 1/4 (for finite Fisher information), so the admitted parameter space includes gamma in (-1,-1/2], a regime where GEV-family ML estimators are known to be non-regular and not asymptotically normal with the usual inverse-information covariance. Section 4's simulations only examine gamma in {-0.2, 0, 0.2}, all safely in the regular region, so they provide no empirical support for the problematic range. The application has fitted gamma = -0.155, which is plausibly regular, so the headline Itajai finding may well be fine; but the paper's general claim to 'rigorous identification' is not established for the parameter space it claims. The lack of a simulation study for the diagnostic itself, rather than for MLE consistency, and the absence of code compound this, but the regularity gap is the primary issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops local influence diagnostics for an extreme-value Birnbaum-Saunders (EVBS) regression model. It derives the score function and Hessian of the log-likelihood, proposes conformal normal curvature diagnostics under case-weighting, response-perturbation, and explanatory-variable-perturbation schemes, examines MLE performance through simulations for three gamma values, and applies the case-weighting diagnostic to monthly maximum wind gust data from Itajaí, Brazil. The analysis identifies observation 82 as strongly influential; removing it changes the tail-shape estimate gamma from -0.1551 to -0.2694, a change of about 73.67%, and the observation corresponds to a documented severe storm with casualties.","tokens_in":28379,"tokens_out":13631,"duration_ms":135016,"significance":"If the derivation is correct, the paper would provide a useful diagnostic toolkit for a regression model that is increasingly used in environmental and physical applications. The case-weighting application is a genuine strength: the flagged observation is supported by a large change in gamma, and the documented storm is used only as external confirmation rather than as a tuning target, so the analysis is not circular. The paper is also clearly written in its exposition of Cook/Poon-Poon influence methodology. However, the broader significance is currently limited by three issues: the claimed regularity conditions cover a non-regular parameter range without proof, the response- and explanatory-perturbation formulas in the general gamma case appear to contain sign and ratio errors, and the simulation study does not evaluate the diagnostic itself. These issues do not necessarily invalidate the case-weighting application, whose fitted gamma lies in the regular region, but they do prevent the paper from supporting its general claim to rigorous identification of influential observations.","major_comments":[{"comment":"The diagnostic machinery presupposes standard MLE asymptotics, but the paper explicitly declines to establish them. Section 3.1 states that the support of the log-EVBS distribution depends on the parameters and that verifying the regularity conditions 'is a problem that deserves to be investigated, but it is far from the objective of this work.' Section 5.2 then assumes gamma > -1 and gamma < 1/4. This admitted parameter range includes gamma in (-1, -1/2], the known non-regular regime for GEV-type likelihoods, where the MLE is not asymptotically normal at the usual n^{-1/2} rate and the inverse observed Hessian does not have its customary covariance interpretation. The Poon-Poon threshold q/sqrt(n) and the curvature formulas in Eqs. (21) and (24) therefore lack their nominal justification over the stated parameter space. The Section 4 simulations cover only gamma in {-0.2, 0, 0.2}, all in the regular region, so they provide no empirical support for the problematic range. The application's fitted gamma = -0.155 is plausibly regular, so the headline Itajaí finding may well be sound, but the paper's general claim to 'rigorous identification' is not established as written. The authors should either restrict the claimed validity to the regular range (e.g., gamma > -1/2 under the cited GEV conditions), prove the needed regularity for log-EVBS, or provide simulation evidence covering the non-regular range.","section":null},{"comment":"The response-perturbation formulas for gamma != 0 appear to be incorrect. Differentiating the individual score term dL/dbeta_j from Eq. (17) with respect to omega_i under y_iw = y_i + omega_i s_y gives, at omega = 0, a quantity proportional to (xi_{i2}/xi_{i1})^2 - 1 + (xi_{i2}/(1+gamma*xi_{i2}))*(1+gamma - (1+gamma*xi_{i2})^{-1/gamma}) + (1+gamma)*(xi_{i1}/(1+gamma*xi_{i2}))^2*((1+gamma*xi_{i2})^{-1/gamma} - gamma). Equation (38), however, prints 1 - xi_{i1wr}^2/xi_{i2wr}^2 and gives the opposite signs to the last two terms. The same inversion and sign pattern reappears in Eqs. (39)-(40) and in the explanatory-variable formulas (45)-(47). These two perturbation schemes therefore do not implement the curvature formula (24) as claimed, and any influence classification based on them would not be reliable as printed. The case-weighting formulas in Eqs. (31)-(33) are consistent with the score functions, and since the application uses only case-weighting, this error does not by itself invalidate the Itajaí result, but the paper's general methodological claim covers all three schemes and must be corrected.","section":null},{"comment":"The simulation study evaluates only the bias, RMSE, and coverage probability of the maximum likelihood estimators. The paper's central claim, however, concerns the local influence diagnostic: that the CNC eigenvalues and the aggregate contribution in Eq. (26) reliably identify influential observations. No simulation perturbs a known observation or measures whether the q-influential eigenvectors and the quantities B_j(q) recover the intended observation. Without such a study, the operating characteristics of the diagnostic are unverified even in the regular parameter region. The authors should add a simulation for the diagnostic procedure itself, for example by contaminating one or more observations with known perturbations and reporting the true-positive rate of the proposed threshold, or they should substantially temper the language of 'rigorous identification' in the abstract and conclusion.","section":null}],"minor_comments":[{"comment":"In the gamma = 0 case, the log-likelihood should begin with -n log 2, not +n log 2, because the density is f(y) = (1/2)*xi_1*exp(-xi_2 - exp(-xi_2)). The score functions in Eq. (18) are unaffected, but the displayed log-likelihood is inconsistent with the density in Eq. (6).","section":null},{"comment":"There are several typographical artifacts: 'e' should be 'and' in expressions such as 'i = 0, 1, ..., p e j = 1, 2, ..., n', and 'senh' should be 'sinh' throughout the perturbation formulas.","section":null},{"comment":"In the gamma = 0 explanatory-variable perturbation, the term xi_{j2wc}/xi_{j2wc} appears to be a typo for xi_{j2wc}/xi_{j1wc}; as printed it is identically 1 and cannot be the intended derivative term.","section":null},{"comment":"The symbol q is used both for the dimension of the perturbation vector ('we assume q = n') and for the integer threshold in the q-influential criterion. This overloading should be resolved to avoid confusion.","section":null},{"comment":"The manuscript provides a link to the data but not to the R code used for the simulations and the application. Given the complexity of the Hessian and perturbation formulas, providing code would substantially improve reproducibility and would help readers verify the corrected formulas.","section":null}],"recommendation":"major_revision","confidential_remarks":"The manuscript is salvageable and should not be rejected outright: the case-weighting machinery appears internally coherent, and the Itajaí application is a convincing external validation. The revision, however, is substantial. The response- and explanatory-perturbation formulas in Section 5.2 must be corrected or removed, the regularity claims must be restricted to the range where the MLE asymptotics are actually justified, and the simulation study should evaluate the diagnostic itself. If the authors prefer to refocus the paper on case-weighting only, that would be an acceptable alternative, but the current general framing overstates what is established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a competent specialist paper, not a breakthrough. The genuinely new content is the set of conformal normal curvature formulas for log-EVBS regression under case-weighting, response, and explanatory-variable perturbations; those expressions are not in Leiva et al. (2016) or the other cited work. The case-weighting application is the strongest part: it flags observation 82, the April 2017 Itajaí storm, and removing it changes the tail estimate by roughly 74%, a clean, externally documented demonstration.\n\nWhat it does well: the simulation study is real—5,000 replications, three scenarios, analytic Hessian—and supports consistency of the joint MLE in the settings examined. The moment conditions in Section 2.2 are correctly argued. Residual checks are standard but adequate.\n\nMain soft spot: the diagnostic machinery inherits an unproved regularity premise. Section 3.1 explicitly says the support depends on parameters and that verifying standard MLE conditions is far from the objective; the authors maintain GEV-family assumptions. That is a real gap because the curvature formulas and the Poon-Poon q/√n threshold rely on the inverse observed Hessian being a valid covariance estimator. The paper's own assumptions allow γ in (-1,-1/2], where GEV ML is known to be non-regular, and the simulations only cover γ in {-0.2, 0, 0.2}. The fitted γ is -0.155, safely in the regular region, so the headline finding is probably fine—but the general claim to rigorous identification is not established for the stated parameter space. Flag it, don't kill the paper over it.\n\nAlso: Section 5.2 has apparent typos in the response and explanatory perturbation formulas for γ≠0—the ξ1/ξ2 ratios are inverted in places—and the γ=0 log-likelihood in equation (14) has a sign error. No code is shipped, though the data are public. The q-influence threshold and scale factors s_y, s_x are choices; they are not fully justified but are standard.\n\nVerdict: for readers working on EVBS or Birnbaum-Saunders regression diagnostics, this deserves a serious referee. I'd send it out, with a request to fix the typos and either narrow the generality claim or address the non-regular regime. It is not a paper that changes how extreme events are modeled, but it is a solid toolkit for a specific model class.","headline":"Solid subfield contribution: new local-influence curvature formulas for log-EVBS regression plus a convincing case study, but the general diagnostic claim rests on an openly unproved MLE-regularity assumption.","tokens_in":28910,"tokens_out":3258,"would_cite":true,"duration_ms":36319,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G32","62J20","62F12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that conformal-normal-curvature diagnostics, derived in closed form for extreme-value Birnbaum-Saunders regression, identify the single storm that dominates a wind-gust sample; removing that flagged storm shifts the…","keywords":["conformal normal curvature","extreme value theory","Birnbaum-Saunders distribution","local influence diagnostics","regression stability","wind gust extremes","influential observations"],"falsifier":"Simulate 5,000 datasets from the fitted model ($\\alpha \\approx 0.186$, $\\gamma \\approx -0.155$, $n=124$) with no planted anomaly and apply the case-weighting CNC benchmark: if any point crosses the $q/\\sqrt{n}$ threshold in a large fraction of replications, the threshold does not control false positives. A complementary check would run the same diagnostic on a second weather station from the same network that has no documented catastrophic storm; if a single observation is still flagged as clearly influential, the method is responding to ordinary leverage rather than to physical extremity.","tokens_in":27874,"feed_emoji":"🌬️","tokens_out":11564,"duration_ms":119037,"temperature":0.7,"pith_summary":"Extreme-value models are often fitted to data containing rare, physically extreme events, so a single anomalous observation can quietly control the fitted tail behavior. The paper develops a local-influence diagnostic for the extreme-value Birnbaum-Saunders (EVBS) regression model, built from the conformal normal curvature of the log-likelihood, that identifies which observations disproportionately move the parameter estimates. In an application to monthly maximum wind gusts from a Brazilian weather station, the method flags a single documented catastrophic storm as highly influential; deleting that point changes the tail-shape estimate $\\hat{\\gamma}$ from $-0.155$ to $-0.269$, a swing of about 73.67%, while leaving the regression coefficients nearly unchanged. The authors argue that this provides a practical stability check for researchers who use EVBS or related extreme-value regressions to quantify wind, weather, and climate risk.","feed_headline":"One documented storm moves a wind-gust tail estimate by 74 percent","feed_subtitle":"Local curvature diagnostics flag the April 2017 storm and quantify how one point reshapes tail risk.","key_machinery":"The central object is the conformal normal curvature $B_l$ of the likelihood-displacement surface, a normalized curvature that lies in $[0,1]$ and is invariant under conformal reparameterization. The diagnostic reduces to an eigenvalue problem on the matrix $\\Delta^\\top(-\\ddot{L}^{-1})\\Delta$, where $\\Delta = (\\partial^2\\ell(\\theta\\mid\\omega)/\\partial\\theta_i\\partial\\omega_j)$ is the perturbation matrix and $\\ddot{L}$ is the Hessian of the log-likelihood at the maximum-likelihood estimate; eigenvectors of this matrix whose normalized eigenvalues satisfy $\\lambda_i^* \\geq q/\\sqrt{n}$ are declared $q$-influential. The paper derives $\\Delta$ explicitly for case weighting, response perturbation, and explanatory-variable perturbation, and uses an aggregate contribution statistic to rank how much each individual observation contributes to the influential directions.","core_discovery":"The central claim is that local influence diagnostics can be derived in closed form for the log-EVBS regression model and that, when applied through the conformal normal curvature of the likelihood surface, they reliably separate observations that genuinely drive the fit from those that do not. For three perturbation schemes—case weighting, additive perturbation of the response, and additive perturbation of a covariate—the paper computes the perturbation matrix $\\Delta$ whose entries are second derivatives of the perturbed log-likelihood, and combines it with the inverse Hessian to obtain normalized curvatures whose eigenvectors point along directions of maximum sensitivity. On 124 monthly maximum wind-gust values, the case-weighting scheme identifies observation #82, recorded on 26 April 2017, as the only $q$-influential direction at the $q=7$ level; that date corresponds to a severe storm documented in local emergency reports. Removing that observation changes the estimate of the tail-shape parameter $\\gamma$ by $-73.67\\%$, while $\\beta_0$, $\\beta_1$, and $\\alpha$ move by only a few percent, which the paper presents as evidence that the diagnostic pinpoints the physical event that dominates the model's tail behavior.","pith_inferences":["The $q/\\sqrt{n}$ threshold is a geometric reference value, not a calibrated false-discovery rate; a natural extension would be to simulate from the fitted model and measure how often an ordinary observation is classified as influential under each perturbation scheme.","Because $\\gamma$ controls tail weight, the 73.67% shift implies that estimated return levels for extreme wind speeds would move substantially when the storm is included versus excluded; the paper does not report those return-level changes, but they follow directly from the reported parameter swing.","The same $\\Delta$-matrix construction should transfer to nonstationary GEV regressions with covariates on all three parameters, giving extreme-value practitioners a screening tool for time series with known catastrophic events."],"forward_implications":["Fitting an EVBS regression no longer requires case-deletion recomputation: the closed-form $\\Delta$ matrices let a practitioner screen all $n$ observations for influence in a single pass.","A single verified extreme event can dominate the fitted tail: removing the flagged Itajaí storm changes $\\hat{\\gamma}$ by roughly 74%, so tail-shape and return-level conclusions drawn from the full sample hinge on that one point.","The CNC ranking separates physically meaningful influence from marginal influence, because observation #82 meets the $q/\\sqrt{n}$ reference threshold while the marginally flagged observation does not produce a comparable parameter shift.","The same diagnostic structure applies to any future dataset fit with the EVBS regression, and to new perturbation schemes built from the same log-likelihood, because the $\\Delta$ and Hessian expressions are scheme-specific but transferable."],"supporting_citations":[{"why":"Supplies the original local-influence methodology in which the normal curvature of the likelihood displacement measures sensitivity to small perturbations.","marker":"[26]"},{"why":"Provides the conformal normal curvature, its $[0,1]$ normalization, and the $q/\\sqrt{n}$ reference values used to declare directions influential.","marker":"[28]"},{"why":"Introduces the EVBS regression model and the log-EVBS error distribution that the paper's likelihood, score, and Hessian expressions build on.","marker":"[24]"},{"why":"Defines the EVBS distribution as an extreme-value version of the Birnbaum-Saunders distribution, supplying the transformation that links it to the GEV family.","marker":"[23]"},{"why":"One of the regularity results invoked to justify consistency and asymptotic normality of the maximum-likelihood estimator despite the parameter-dependent support of the GEV distribution.","marker":"[38]"},{"why":"Extends maximum-likelihood regularity for the generalized extreme-value distribution, cited alongside [38] and [40] to support the asymptotic assumptions behind the curvature formulas.","marker":"[39]"},{"why":"Provides block-maxima maximum-likelihood regularity results that the paper assumes carry over to the log-EVBS regression setting.","marker":"[40]"},{"why":"Supplies the assumption $\\alpha \\le 2$ that the paper adopts to avoid multiple solutions of the maximum-likelihood equations.","marker":"[41]"},{"why":"Defines the aggregate contribution statistic used to rank how much each observation contributes to the influential eigenvectors.","marker":"[42]"}],"fun_headline_variants":["One storm moves wind-gust tail estimate 74%","Diagnostics flag April 2017 storm's 74% tail shift","Influence analysis pinpoints storm altering tail shape","Wind-gust model: single storm changes tail estimate 74%","Curvature method finds storm that sways tail fit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire diagnostic machinery assumes the maximum-likelihood estimator is consistent and asymptotically normal with an invertible Hessian, even though the support of the log-EVBS distribution depends on the parameter values; Section 3.1 explicitly sets this regularity question aside, so if that premise fails the inverse Hessian and the curvature thresholds lose their nominal interpretation.","fun_headline_variants_meta":{"raw":{"variants":["One storm moves wind-gust tail estimate 74%","Diagnostics flag April 2017 storm's 74% tail shift","Influence analysis pinpoints storm altering tail shape","Wind-gust model: single storm changes tail estimate 74%","Curvature method finds storm that sways tail fit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000235,"raw_usage":{"total_tokens":1516,"prompt_tokens":979,"completion_tokens":537,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":453}},"tokens_in":595,"tokens_out":537,"duration_ms":6297,"temperature":1.0,"reasoning_tokens":453,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:14:30.376791+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate 5,000 datasets from the fitted model ($\\alpha \\approx 0.186$, $\\gamma \\approx -0.155$, $n=124$) with no planted anomaly and apply the case-weighting CNC benchmark: if any point crosses the $q/\\sqrt{n}$ threshold in a large fraction of replications, the threshold does not control false positives. A complementary check would run the same diagnostic on a second weather station from the same network that has no documented catastrophic storm; if a single observation is still flagged as clearly influential, the method is responding to ordinary leverage rather than to physical extremity.","supporting_citations":[{"cited_title":"Dennis Cook","cited_arxiv_id":null,"evidence_quote":"Supplies the original local-influence methodology in which the normal curvature of the likelihood displacement measures sensitivity to small perturbations."},{"cited_title":"Conformal normal curvature and assessment of local influence","cited_arxiv_id":null,"evidence_quote":"Provides the conformal normal curvature, its $[0,1]$ normalization, and the $q/\\sqrt{n}$ reference values used to declare directions influential."},{"cited_title":"Ivette Gomes, and Camilo Lillo","cited_arxiv_id":null,"evidence_quote":"Introduces the EVBS regression model and the log-EVBS error distribution that the paper's likelihood, score, and Hessian expressions build on."},{"cited_title":"Ferreira, M","cited_arxiv_id":null,"evidence_quote":"Defines the EVBS distribution as an extreme-value version of the Birnbaum-Saunders distribution, supplying the transformation that links it to the GEV family."},{"cited_title":"Existence and consistency of the maximum likelihood estimators for the extreme value index within the block maxima framework.Bernoulli, 21(1):420–436, 2015","cited_arxiv_id":null,"evidence_quote":"One of the regularity results invoked to justify consistency and asymptotic normality of the maximum-likelihood estimator despite the parameter-dependent support of the GEV distribution."},{"cited_title":"Bucher and J","cited_arxiv_id":null,"evidence_quote":"Extends maximum-likelihood regularity for the generalized extreme-value distribution, cited alongside [38] and [40] to support the asymptotic assumptions behind the curvature formulas."},{"cited_title":"Maximum likelihood estimators based on the block maxima method","cited_arxiv_id":null,"evidence_quote":"Provides block-maxima maximum-likelihood regularity results that the paper assumes carry over to the log-EVBS regression setting."},{"cited_title":"Rieck and Jerry R","cited_arxiv_id":null,"evidence_quote":"Supplies the assumption $\\alpha \\le 2$ that the paper adopts to avoid multiple solutions of the maximum-likelihood equations."},{"cited_title":"Application of Elementary Differential Geometry to Influence Analysis","cited_arxiv_id":null,"evidence_quote":"Defines the aggregate contribution statistic used to rank how much each observation contributes to the influential eigenvectors."}],"review_version":1}