On the study of the Beran estimator for generalized censoring indicators

Along with the analysis of time-to-event data, it is common to assume that only partial information is given at hand. In the presence of right-censored data with covariates, the conditional Kaplan-Meier estimator (also referred as the Beran estimator) is known to propose a consistent estimate for the lifetimes conditional survival function. However, a necessary condition is the clear knowledge of whether each individual is censored or not, although, this information might be incomplete or even totally absent in practice. We thus propose a study on the Beran estimator when the censoring indicator is not clearly specified. From this, we provide a new estimator for the conditional survival function and establish its asymptotic normality under mild conditions. We further study the supervised learning problem where the conditional survival function is to be predicted with no censorship indicators. To this aim, we investigate various approaches estimating the conditional expectation for the censoring indicator. Along with the theoretical results, we illustrate how the estimators work for small samples by means of a simulation study and show their practical applicability with the analysis of synthetic data and the study of real data for the prognosis of monoclonal gammopathy.