Asymptotic properties of nonparametric estimation based on partly interval-censored data

We study asymptotic properties of the nonparametric maximum likelihood estimator (NPMLE) of a distribution function based on partly interval-censored data in which the exact values of some failure times are observed in addition to interval-censored observations. It is shown that the NPMLE converges weakly to a mean zero Gaussian process whose covariance function is determined by a Fredholm integral equation. Simulations are conducted to demonstrate that the NPMLE based on all the observations substantially outperforms the empirical distribution function, using only the fully observed observations, in terms of the mean square error. It is also shown that the nonparametric bootstrap estimator of the distribution function is first order consistent, which provides asymptotic justification for the use of bootstrap to construct confidence bands for the unknown distribution function.