Estimating a Distribution Function for Censored Time Series Data

Consider a long term study, where a series of dependent and possibly censored failure times is observed. Suppose that the failure times have a common marginal distribution function, but they exhibit a mode of time series structure such as [alpha]-mixing. The inference on the marginal distribution function is of interest to us. The main results of this article show that, under some regularity conditions, the Kaplan-Meier estimator enjoys uniform consistency with rates, and a stochastic process generated by the Kaplan-Meier estimator converges weakly to a certain Gaussian process with a specified covariance structure. Finally, an estimator of the limiting variance of the Kaplan-Meier estimator is proposed and its consistency is established.