The problem of estimating the mean of an exponential distribution is studied, when the data available are a random sample of time-censoredrth-order statistics. Examples for such empirical situations are cited. Maximum likelihood estimators (MLE's) and moment-equation estimators (MEE's) are studied. Theoretical derivations are provided for the large sample variances and distributions of these estimators. The efficiency of the MEE compared to the MLE is studied.