{
  "abstract": "Parameter estimation and classification in binary logistic regression deteriorate severely when the number of minority class observations ($n_1$) is smaller than the feature dimension ($p$). Under such rank-deficient rarity ($n_1 < p$), maximum likelihood estimators suffer from quasi-complete separation, penalized likelihood methods exhibit severe intercept bias, and convex minority oversampling techniques such as SMOTE collapse into an $(n_1 - 1)$-dimensional affine subspace. In this paper, we propose a novel \\textit{Self-Calibrated Zero-Complement} estimator that shifts the paradigm from minority interpolation to majority fringe extrapolation. Grounded in Stein's Lemma, we prove that the conditional expectation shift of the abundant majority class ($Y=0$) induced by logistic truncation uniquely identifies the true hyperplane normal $\\boldsymbolβ$ after Mahalanobis whitening. We further establish that extrapolating synthetic minority instances outward from the high-quantile density fringe of the majority distribution along the conjugate direction preserves full-rank covariance regularity ($\\mathrm{rank} = p$). To accommodate both semi-supervised environments (with an unlabeled background mean) and strictly supervised subsamples, we introduce a hybrid estimator incorporating the intrinsic third-moment skewness of the majority class alongside a location-shift-invariant zero-boundary intercept calibration. Extensive Monte Carlo simulations and real-world benchmark applications (Wisconsin Diagnostic Breast Cancer and Banknote Authentication) under extreme rarity ($n_1 \\in \\{3, 5\\}$) demonstrate that our approach prevents parameter divergence, recovers the oracle hyperplane with high fidelity, and substantially outperforms Firth's regression and SMOTE in F1-score and estimation stability.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39194",
  "author_total": 1,
  "authors": [
    {
      "name": "Chongsun Park"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6st1626hyf150.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39194",
  "fulltext_url": null,
  "id": "6st1626hyf150",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39194",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.ML"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39194",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6st1626hyf150.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:41Z",
      "native_id": "2610.39194",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39194"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ME",
      "subfield": "statistics"
    }
  ],
  "title": "Zero-Complement Extrapolation: Parameter Recovery and Classification in Severely Imbalanced Logistic Regression under Rank-Deficient Regimes",
  "title_text": "Zero-Complement Extrapolation: Parameter Recovery and Classification in Severely Imbalanced Logistic Regression under Rank-Deficient Regimes",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/6st1626hyf150",
  "venue": "arXiv",
  "versions": []
}