Prentice (1986) introduced the case-cohort design to reduce the cost and necessary manpower in conducting large cohort studies that arise from processing all information, especially those requiring lab analyses and coding of subject-maintained diaries, into measurable covariates. In this design all variables are measured on everyone in a randomly selected subcohort, but only on those experiencing the event of interest in the remaining non-subcohort. The goal in this presentation is the estimation of the group-specific survival functions when group membership is only partially known, as from case-cohort data. The nonparametric maximum likelihood estimator is derived along with some smoothed variations that require fewer assumptions. Consistency and asymptotic normality are considered. The small-sample behavior is investigated through simulation. A SEER prostate cancer data set is used to exemplify the methods.