Asymptotic theory for estimators under random censorship

The product limit estimator {Mathematical expression} of an unknown distribution F is represented as a U-statistic plus an error of the order o(1/n). Using this, the maximum likelihood estimator of the specific risk rate in the time interval [0, M], is shown to admit a two term Edgeworth expansion. This risk rate for a specific cause of death is defined as the ratio of the probability of death, due to that particular cause, in the time interval [0, M], to the mean life time of an individual up to that time point M. Similar expansions for the bootstrapped statistics are used to show that the bootstrap distribution, of the studentized estimator of the risk rate, approximates the sampling distribution better than the corresponding normal distribution.