Diaconis–Ylvisaker prior penalized likelihood for $ p/n\to\kappa\in(0,1) $ logistic regression
研究了高维逻辑回归中Diaconis–Ylvisaker先验惩罚似然估计量的行为，当协变量数与样本数之比趋于常数κ∈(0,1)时，该估计量始终存在且可通过标准最大似然方法计算，并提出了调整后的Z统计量和惩罚似然比统计量，适用于最大似然失效的情形。
Summary We characterize the behaviour of the maximum Diaconis–Ylvisaker prior penalized likelihood estimator in high-dimensional logistic regression, where the number of covariates is a fraction $ \kappa\in(0,1) $ of the number of observations $ n $, as $ n\to\infty $. We construct a rescaled estimator with zero asymptotic aggregate bias, and define adjusted $ Z $-statistics and rescaled penalized likelihood ratio statistics that exhibit the typical null asymptotic distributions, when the covariates are independent multivariate normal with an arbitrary covariance matrix and the linear predictor has asymptotic variance $ \gamma^{2} $. While the maximum likelihood estimate asymptotically exists only for a narrow range of $ (\kappa,\gamma) $ values, the maximum Diaconis–Ylvisaker prior penalized likelihood estimate always exists and can be computed directly using standard maximum likelihood routines. Thus, our asymptotic results extend to $ (\kappa,\gamma) $ values for which the maximum likelihood framework breaks down, with no additional implementation or computational cost. We study the estimator’s shrinkage properties, compare the proposed estimation and inference procedures with alternatives that also accommodate proportional asymptotics, and formulate a conjecture, supported by strong empirical evidence, that extends our results to models including an intercept parameter. Finally, we propose estimation methods for all unknown constants involved in our procedures and demonstrate the theoretical advances through extensive simulation studies and the analysis of digit recognition data.