Formalize the proportional-limit theory at the maximum-likelihood endpoint

Establish the proportional-limit behavior of logistic-regression maximum-likelihood estimates with a jointly fitted non-negligible intercept under the Gaussian design considered in the paper, thereby formally covering the endpoint case α=1 of the Diaconis–Ylvisaker-penalized formulation.

Background

The paper analyzes maximum Diaconis–Ylvisaker penalized likelihood (MDYPL) for logistic regression with a fitted intercept and a nonzero prior slope in the proportional regime. The regularization parameter satisfies α∈(0,1), while α=1 corresponds to the unpenalized maximum-likelihood estimator.

The authors note that existing work conjectured behavior for maximum-likelihood logistic regression with a non-negligible fitted intercept that agrees with the MDYPL characterization proved in the paper. However, the present analysis formally excludes α=1, leaving the endpoint case without a proof within this framework.