Penalized and Shrinkage Estimation in the Cox Proportional Hazards Model

Abstract

This article considers the shrinkage estimation procedure in the Cox's proportional hazards regression model when it is suspected that some of the parameters may be restricted to a subspace. We have developed the statistical properties of the shrinkage estimators including asymptotic distributional biases and risks. The shrinkage estimators have much higher relative efficiency than the classical estimator, furthermore, we consider two penalty estimators—the LASSO and adaptive LASSO—and compare their relative performance with that of the shrinkage estimators numerically. A Monte Carlo simulation experiment is conducted for different combinations of irrelevant predictors and the performance of each estimator is evaluated in terms of simulated mean squared error. Simulation study shows that the shrinkage estimators are comparable to the penalty estimators when the number of irrelevant predictors in the model is relatively large. The shrinkage and penalty methods are applied to two real data sets to illustrate the usefulness of the procedures in practice.

Keywords:

• Adaptive LASSO
• Asymptotic distributional bias and risk
• Cox's proportional hazards model
• LASSO
• Likelihood ratio test
• Monte Carlo simulation
• Shrinkage estimators

Mathematics Subject Classification:

• Primary 62N01, 62N02
• Secondary 62N03

Acknowledgments

We would like to thank all the referees, the Editor, and the Associate Editor for their valuable suggestions on the revision of this article.