{"id":"80b86790-32e7-49cd-a4cb-c53224e5b414","arxiv_id":"1908.00462","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":5,"one_line_summary":"Finite-sample breakdown points of the median, Hodges-Lehmann, MAD and Shamos estimators are derived in closed form, and Monte Carlo unbiasing factors and relative efficiencies are tabulated for sample sizes up to 100.","lead":"This paper gives exact formulas for the sample sizes at which common robust estimators such as the median, Hodges-Lehmann, MAD, and Shamos stop tolerating corrupted data, plus Monte Carlo tables of their bias and efficiency for samples up to 100. It also offers fitted formulas to correct the bias of the MAD and Shamos estimators for larger samples, demonstrated in a quality control chart example.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the closed-form breakdown formulas are correct under the paper's explicitly chosen replacement-breakdown definition.","rationale":"The reader's conditional verdict is reasonable, and this stress-test pass does not find a load-bearing flaw in the central claim. The finite-sample breakdown formulas are derived correctly under the replacement-breakdown definition that the paper explicitly adopts. The convention-dependence flagged by the reader is real but is disclosed and discussed in the manuscript, so it does not make the reported formulas wrong; it only means the numbers should not be quoted without the definitional qualification. The simulation-based results, while lacking formal error bars, are ancillary to the central breakdown-point claim, and the extrapolated unbiasing factors are clearly labeled as empirical fits. Overall, no correction to the reader's conditional verdict is needed.","tokens_in":25862,"tokens_out":23986,"duration_ms":242836,"concrete_test":"Implement an exact checker for n = 2,...,20: enumerate all subsets of size m, replace those observations with very large values (using distinct large values for scale estimators), and verify that the estimator bias is finite exactly when m is at most the claimed threshold for each estimator; compare the resulting breakdown points with Table 2 and equations (2), (4), (5), and (6).","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the derivations in Section 2, the closed-form finite-sample breakdown formulas (2), (4), (5), and (6) are internally correct under the paper's Definition 2. The count argument is valid: for a median of N terms, bias remains finite exactly while the number of corrupted terms is at most floor((N-1)/2), and the formulas count corrupted Walsh averages or pairwise differences correctly. Spot checks for n = 2 through 10 agree with Table 2. The main caveat is the choice of breakdown convention: under the alternative Hettmansperger-McKean definition cited in the same section, the reported values would differ, for example, the median at n = 3 would be 2/3 instead of 1/3. However, the authors explicitly state that they follow Equation (1), cite the alternative, and explain why. This is a documented convention choice, not an internal error. The Monte Carlo and least-squares material concerns secondary claims about bias and efficiency, and it does not undermine the central breakdown-point formulas.","agreement_with_reader":"partial"},"referee_report":null,"author_rebuttal":null,"desk_editor":null,"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-14T15:54:47.907955+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}