Empirical and kernel estimation of covariate distribution conditional on survival time

In biomedical research there is often interest in describing covariate distributions given different survival groups. This is not immediately available due to censoring. In this paper we develop an empirical estimate of the conditional covariate distribution under the proportional hazards regression model. We show that it converges weakly to a Gaussian process and provide its variance estimate. We then apply kernel smoothing to obtain an estimate of the corresponding density function. The density estimate is consistent and has the same rate of convergence as the classical kernel density estimator. We have developed an R package to implement our methodology, which is demonstrated through the Mayo Clinic primary biliary cirrhosis data.