Zero-Complement Extrapolation: Parameter Recovery and Classification in Severely Imbalanced Logistic Regression under Rank-Deficient Regimes

Chongsun Park

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…