The proportional hazards regression with a censored covariate

We consider the problem of inference on the regression coefficient in the Cox's proportional hazards regression model with a censored covariate. The relative risk function depends on the baseline hazard as well as the distribution of the covariate. Since the covariate may not be observed due to censoring, we propose a method to empirically estimate the relative risk function based on the uncensored covariate data and derive a partial likelihood function using the estimated relative risk function. This approach enables us to use all possible information contained in the censored covariate data. Asymptotic properties of the proposed estimator are derived, and its efficiency is assessed for exponential failure times using an exponential relative risk function through simulation studies.