The inverse censoring weighted approach for estimation of survival functions from left and right censored data

We propose an inverse censoring weighted estimator of a survival function when the data are doubly censored but the left censoring variable is always observed. The proposed estimator reduces to the standard inverse censoring weighted estimator for right censored data, where there is no left censoring, and in that sense may be viewed as an extension of the latter estimator to the special double censoring scenario considered in this paper. However, the equivalence exhibited by the Kaplan–Meier and inverse censoring weighted estimators in the case of right censoring does not apply any more for the scenario studied here. Specifically, a Kaplan–Meier approach based on a modified risk set and the proposed inverse censoring approach lead to different estimators. Furthermore, when both censoring variables are always observed, as in the case of an AIDS clinical trial data set analyzed in this paper, an alternative inverse censoring weighted estimator can be computed using the additional available censoring information. We present the results of a numerical comparison study between the Kaplan–Meier type and the two inverse censoring weighted estimators.