{"paper":{"title":"Testing axial symmetry around an unspecified direction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Axial symmetry about an unknown direction reduces to testing a finite set of eigenvector candidates using projected Kolmogorov-Smirnov statistics.","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Alejandro Cholaquidis, Juan Cuesta-Albertos, Manuel Hern\\'andez-Banadik, Ricardo Fraiman","submitted_at":"2026-04-10T13:37:10Z","abstract_excerpt":"We consider the problem of testing whether a multivariate distribution is axially symmetric about some unknown direction. Under a simple-spectrum assumption on the covariance matrix, any symmetry axis must coincide with an eigenvector of the covariance matrix, so the problem reduces to testing a finite set of candidate directions. For each candidate direction, we construct a Kolmogorov--Smirnov-type statistic based on projected data and sample splitting. We derive its asymptotic distribution in a triangular-array framework and establish bootstrap validity under suitable regularity conditions. "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This leads to a feasible testing procedure for axial symmetry when the symmetry direction is unspecified.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Under a simple-spectrum assumption on the covariance matrix, any symmetry axis must coincide with an eigenvector of the covariance matrix.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A Kolmogorov-Smirnov-type test for axial symmetry about an unspecified direction that reduces to testing a finite set of covariance eigenvectors under a simple-spectrum assumption, with derived asymptotics and bootstrap validity.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Axial symmetry about an unknown direction reduces to testing a finite set of eigenvector candidates using projected Kolmogorov-Smirnov statistics.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c4115f41388295e9e40f94e86efab035fc9a46cf846a03fa4e34efbac14d21e5"},"source":{"id":"2604.09317","kind":"arxiv","version":2},"verdict":{"id":"f1d58206-2c4a-4763-b40a-6661d765a5f9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T16:56:03.890709Z","strongest_claim":"This leads to a feasible testing procedure for axial symmetry when the symmetry direction is unspecified.","one_line_summary":"A Kolmogorov-Smirnov-type test for axial symmetry about an unspecified direction that reduces to testing a finite set of covariance eigenvectors under a simple-spectrum assumption, with derived asymptotics and bootstrap validity.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Under a simple-spectrum assumption on the covariance matrix, any symmetry axis must coincide with an eigenvector of the covariance matrix.","pith_extraction_headline":"Axial symmetry about an unknown direction reduces to testing a finite set of eigenvector candidates using projected Kolmogorov-Smirnov statistics."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.09317/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}