{"paper":{"title":"The Threshold Breakdown Point","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The threshold breakdown point is the smallest contamination fraction that forces an estimator past a chosen deviation level.","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Marco Avella Medina, Tianjun Ke","submitted_at":"2026-05-05T21:36:45Z","abstract_excerpt":"We introduce a novel approach to finite sample robustness that avoids the pessimism of traditional breakdown analyses. We define the threshold breakdown point, the smallest contamination fraction needed to induce a prescribed deviation, and the finite sample m-sensitivity, the worst-case deviation that an estimator can incur after m observations are contaminated. We derive these measures for commonly used M-estimators, their standard errors and related test statistics. This allows us to extend the decision breakdown point of Zhang (1996) to obtain general breakdown characterizations for hypoth"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We define the threshold breakdown point, the smallest contamination fraction needed to induce a prescribed deviation, and the finite sample m-sensitivity, the worst-case deviation that an estimator can incur after m observations are contaminated. We derive these measures for commonly used M-estimators, their standard errors and related test statistics.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The derivations and extensions assume that M-estimators satisfy the regularity conditions needed for explicit breakdown calculations and that the contamination model permits well-defined worst-case deviations; the inferential results further assume standard asymptotic conditions for consistency, normality, and bootstrap validity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Introduces threshold breakdown point and m-sensitivity as new finite-sample robustness measures for M-estimators and tests, with consistency, asymptotic normality, and multiplier bootstrap inference.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The threshold breakdown point is the smallest contamination fraction that forces an estimator past a chosen deviation level.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"514749789ba997bff077ade78e47d4aee23a0890521c18ddecc0236bbdf43c3a"},"source":{"id":"2605.04317","kind":"arxiv","version":2},"verdict":{"id":"1a18613f-b904-4e76-9f21-364a1c62ce53","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T16:46:37.496161Z","strongest_claim":"We define the threshold breakdown point, the smallest contamination fraction needed to induce a prescribed deviation, and the finite sample m-sensitivity, the worst-case deviation that an estimator can incur after m observations are contaminated. We derive these measures for commonly used M-estimators, their standard errors and related test statistics.","one_line_summary":"Introduces threshold breakdown point and m-sensitivity as new finite-sample robustness measures for M-estimators and tests, with consistency, asymptotic normality, and multiplier bootstrap inference.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The derivations and extensions assume that M-estimators satisfy the regularity conditions needed for explicit breakdown calculations and that the contamination model permits well-defined worst-case deviations; the inferential results further assume standard asymptotic conditions for consistency, normality, and bootstrap validity.","pith_extraction_headline":"The threshold breakdown point is the smallest contamination fraction that forces an estimator past a chosen deviation level."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04317/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:31:20.591183Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:31:35.198353Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"81e3247267e08cd8f7b3e105e08504e9ed3b4880a8b89581a696be63c6cf2cb4"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}