AI 中文总结
本文为带拟合截距的Diaconis-Ylvisaker惩罚逻辑回归建立比例极限下的估计量渐近理论，给出斜率极限律、检验统计量分布及可行推断方法。
AI 中文摘要
本文针对逻辑回归中联合拟合截距和非零先验斜率方向的最大Diaconis-Ylvisaker先验惩罚似然，在比例极限机制下发展了估计量层面的渐近理论。对于N(0_p, p^{-1}I_p)高斯协变量且p/n→κ∈(0,1)的情形，条件凸高斯极小-极大分析导出了拟合截距的几乎必然收敛以及斜率估计量的伪Lipschitz经验律。该估计量层面的律给出了样本外得分、分类误差、最优阈值和oracle校准的渐近极限。它还在各向同性高斯协变量下导出了oracle调整的固定块Z统计量，并提供了建立惩罚似然比检验统计量极限分布的主要成分，同时确定了恢复名义卡方分布的重新缩放。我们将这些结果通过仿射中心化和白化扩展到具有任意确定性均值和正定协方差的高斯设计，并讨论了次高斯协变量的扩展。最后，我们提出了一致响应矩估计量，用于估计进入状态方程（控制斜率极限律且可行推断所需）的oracle参数。
英文摘要
This paper develops estimator-level asymptotic theory for maximum Diaconis-Ylvisaker prior penalised likelihood for logistic regression with a jointly fitted intercept and nonzero prior slope direction in the proportional-limit regime. For $\mathrm{N}(\mathbf{0}_p, p^{-1}\mathbf{I}_p)$ Gaussian covariates and $p/n\toκ\in(0,1)$, a conditional convex Gaussian min--max analysis yields almost-sure convergence of the fitted intercept and a pseudo-Lipschitz empirical law for the slope estimator. This estimator-level law gives asymptotic limits for out-of-sample scores, classification error, optimal thresholding and oracle calibration. It also yields oracle-adjusted fixed-block $Z$-statistics under isotropic Gaussian covariates, and yields the main ingredient in establishing the limiting distribution of the penalised likelihood-ratio test statistic and identifies the rescaling to recover the nominal chi-square distribution. We extend these results to Gaussian designs with arbitrary deterministic mean and positive-definite covariance via affine centering and whitening and discuss extensions to subgaussian covariates. Finally, we propose a consistent response-moment estimator of the oracle parameters entering the state equations that govern the slope limiting law and are required for feasible inference.
