Non-parametric inference of a failure time distribution when the failure times are estimated

We consider the estimation of a failure time distribution F when, instead of N i.i.d. realizations T1, T2, ..., TN from F, the observations consist of estimates of the Ti. If the Ti could be observed, a natural non-parametric estimator of F would be the Kaplan-Meier estimator. Thus, we examine the properties of the Kaplan-Meier estimator based on the estimates of the Ti. We also consider a weighted Kaplan-Meier estimator which gives more emphasis to those estimated times based on more information. We evaluate the small sample bias and precision of F when the estimated failure times arise from additive or multiplicative error structures. Because this problem has particular application in the study of non-compliance of subjects in clinical trials, we also investigate the bias and precision of the estimators of the distribution function based on a complex error structure that would arise in the non-compliance setting. Here Ti denotes the unobserved time to non-compliance for the ith subject, and is estimated using repeated observations from a laboratory marker whose behaviour is affected by non-compliance. The techniques are illustrated with the results of a recent AIDS clinical trial.