Integrating preliminary test and Stein-type techniques to improve estimation in the time-dependent Cox model

While shrinkage and preliminary test estimation have long been studied in linear and static Cox models, their theoretical integration within models featuring time-dependent covariates has remained unresolved due to the evolving risk set and nonhomogeneous information accumulation inherent in such data. In this study, we develop a unified framework for shrinkage estimation in the time-dependent Cox proportional hazards model, by extending the classical Stein-type theory to a dynamic semiparametric survival setting. Our theoretical analyses reveal that the positive-rule Stein estimator preserves unbiasedness under valid restrictions while adaptively attenuating variance inflation when the restriction is approximately correct, striking a principled balance between efficiency and robustness. A comprehensive Monte Carlo simulation study and an empirical application to the Mayo Clinic primary biliary cirrhosis dataset substantiate the theoretical advantages, demonstrating that the superior estimation strategy achieves substantial efficiency gains relative to both unrestricted and penalized estimators such as adaptive LASSO.