{
  "abstract": "We introduce a distribution-free goodness-of-fit test, termed the omega-1 test, which naturally complements the Kolmogorov--Smirnov test and Cramér--von Mises test and can be viewed as their (piecewise) linear analog. Defined as an $\\mathrm{L}^{1}$-functional of the empirical process, the test statistic improves on balancing sensitivity to localized and diffuse alternatives and gives a robust and interpretable measure of distributional discrepancy, apart from close connections to the Wasserstein 1-distance. For finite samples, we derive a finite-dimensional computational form for the statistic under general conditions, which leads to various explicit formulas for its null distribution. Under mild continuity assumptions, the limiting statistic is distribution-free, with explicit distribution formulas. In composite settings, the statistic is also compatible with the Khmaladze transformation, enabling asymptotically distribution-free testing. The limiting transformed statistic also has an explicit distribution that escapes reliance on intractable compensator processes or purely numerical evaluation. Simulation results indicate rapid convergence of the finite-sample distributions to their limiting counterparts and support the practical applicability of the test.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30074",
  "author_total": 1,
  "authors": [
    {
      "name": "Weixuan Xia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86wjjta7qh5ap.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30074",
  "fulltext_url": null,
  "id": "86wjjta7qh5ap",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30074",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Statistical Distribution Estimation and Applications"
    ]
  ],
  "openalex": "W7204895148",
  "pdf_url": "https://arxiv.org/pdf/2608.30074",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86wjjta7qh5ap.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30074",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30074"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Distribution-free testing of linear type",
  "title_text": "Distribution-free testing of linear type",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:54Z",
  "url": "https://research.draconitic.com/paper/86wjjta7qh5ap",
  "venue": "arXiv",
  "versions": []
}