Proportional-limit asymptotics for Diaconis-Ylvisaker-penalised logistic regression with fitted intercept
Abstract
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 N(0p, p−1Ip) Gaussian covariates and p/n→κ∈(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.
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BibTeX
@misc{sterzinger2026proportionallimit,
title = {{Proportional-limit asymptotics for Diaconis-Ylvisaker-penalised logistic regression with fitted intercept}},
author = {Philipp Sterzinger},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09831},
eprint = {2609.09831},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09831},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Proportional-limit asymptotics for Diaconis-Ylvisaker-penalised logistic regression with fitted intercept
AU - Philipp Sterzinger
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09831
UR - https://arxiv.org/abs/2609.09831
AB - 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 N(0_p, p^−1I_p) Gaussian covariates and p/n→κ∈(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.
ER -