Consistent Estimation in Cox Proportional Hazards Model with Covariate Measurement Errors

The regular maximum partial likelihood estimator is biased when the covariates in the Cox proportional hazards model are measured with error, unless the measurement errors tend to zero. Although several alternative estimators have been proposed, theoretical justifications for them are lacking. We try to fill this gap by showing that the corrected maximum partial likelihood estimator proposed by Nakamura (1992) is consistent and has an asymptotic normal distribution. A consistent estimator of its variance is derived as well. We also show that the corrected baseline hazard estimator proposed by Kong, Huang and Li (1998) is consistent and converges to a Gaussian process. Furthermore, we obtain the estimators for more general measurement error models where the errors are not normally distributed. Simulations are performed to show the accuracy of the variance estimator.