{"paper":{"title":"Pattern-based tests for two-dimensional copulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A functional central limit theorem for pattern frequencies in bivariate rank plots enables nonparametric copula tests.","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"L. Baringhaus, R. Gr\\\"ubel","submitted_at":"2026-05-13T15:57:05Z","abstract_excerpt":"In statistics permutations typically arise in the context of rank plots for two-dimensional data. Such plots can also be interpreted as discrete copulas. In discrete mathematics, typically in the context of the description of large (non-random) objects, two-dimensional copulas appear as limits of permutations and are then known as permutons if the topology refers to the convergence of pattern frequencies. We obtain a functional central limit theorem for such pattern frequencies in the context of two-dimensional random samples. The result serves as the basis for nonparametric goodness-of-fit te"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We obtain a functional central limit theorem for such pattern frequencies in the context of two-dimensional random samples. The result serves as the basis for nonparametric goodness-of-fit tests, for two-sample tests, and for tests of symmetry.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The two-dimensional samples are i.i.d. from a continuous bivariate distribution, permitting the rank plot to be treated as a discrete copula whose pattern frequencies converge in the permuton topology.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A functional central limit theorem for pattern frequencies in 2D samples enables nonparametric goodness-of-fit, two-sample, and symmetry tests for copulas, with bootstrap critical values and parametric examples.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A functional central limit theorem for pattern frequencies in bivariate rank plots enables nonparametric copula tests.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"01e8e0e6cd371a675c174298afc742c4219c553f8682dbdcde2665f867a5564d"},"source":{"id":"2605.13710","kind":"arxiv","version":1},"verdict":{"id":"724ff6f8-d0ee-410d-8111-165f56a2ca75","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:36:36.297905Z","strongest_claim":"We obtain a functional central limit theorem for such pattern frequencies in the context of two-dimensional random samples. The result serves as the basis for nonparametric goodness-of-fit tests, for two-sample tests, and for tests of symmetry.","one_line_summary":"A functional central limit theorem for pattern frequencies in 2D samples enables nonparametric goodness-of-fit, two-sample, and symmetry tests for copulas, with bootstrap critical values and parametric examples.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The two-dimensional samples are i.i.d. from a continuous bivariate distribution, permitting the rank plot to be treated as a discrete copula whose pattern frequencies converge in the permuton topology.","pith_extraction_headline":"A functional central limit theorem for pattern frequencies in bivariate rank plots enables nonparametric copula tests."},"references":{"count":52,"sample":[{"doi":"10.1137/1","year":1971,"title":"More asymmetry yields faster matrix multiplication","work_id":"5ec132ea-8bab-4415-b2af-677e9db16f00","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1080/03610919408813193","year":1994,"title":"Baringhaus, L. (1994). On a modification of the Hoeffding–Blum–Kiefer–Rosenblatt independence criterion. Comm. Statist. Simulation Comput.23683–689. https://doi.org/10.1080/03610919408813193","work_id":"2f6157f4-7736-48f7-b91b-f49804616ecd","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/s0047-259x(03)00079-4","year":2004,"title":"Baringhaus, L., Franz, C. (2004). On a new multivariate two-sample test.J. Multivariate Anal.88190–206. https://doi.org/10.1016/S0047-259X(03)00079-4","work_id":"71fce151-2ad9-4889-9128-6f4bb2ee2881","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"Baringhaus, L., Franz, C. (2010). Rigid motion invariant two-sample tests.Statist. Sinica201333–1361. Tests for copulas 25","work_id":"7d9e5d1b-0a07-4d3a-af0a-91b222d19555","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1214/23-ejs2205","year":2024,"title":"Baringhaus, L., Grübel, R. (2024). Random permutations generated by delay models and estimation of delay distributions.Electron. J. Stat.18167–190. https://doi.org/10.1214/23-EJS2205","work_id":"86f6835c-5f72-43a0-b411-31fdb361c12b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":52,"snapshot_sha256":"28fa5fac22d11952209af170719b0cd53d83a17f94138be7e2c1dfef65b48be6","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"}