Identity for the NPMLE in censored data models

We derive an identity for nonparametric maximum likelihood estimators (NPMLE) and regularized MLEs in censored data models which expresses the standardized maximum likelihood estimators in terms of the standardized empirical process. This identity provides an effective starting point in proving both consistency and efficiency of NPMLE and regularized MLE. The identity and corresponding method for proving efficiency is illustrated for the NPMLE in the univariate right-censored data model, the regularized MLE in the current status data model and for an implicit NPMLE based on a mixture of right-censored and current status data. Furthermore, a general algorithm for estimation of the limiting variance of the NPMLE is provided.