Nonparametric estimation from cross-sectional survival data

In many follow-up studies survival data are often observed according to a cross-sectional sampling scheme. Data of this type are subject to left truncation in addition to the usual right censoring. A number of characteristics and properties of the product-limit estimate, for left-truncated and right-censored data, have been explored and found to be similar to those of the Kaplan–Meier estimate. Under the stationarity assumption, however, it is believed that an alternative estimate has much better efficiency. In this article the conditional maximum likelihood estimate (MLE) property of the product–limit estimate is visited. The non-parametric MLE of the truncation distribution is derived. Use of this estimate includes testing the stationarity assumption, estimating the proportion of truncated data, and other applications in prevalent cohort studies. The analysis of the estimation is based on a “working data” approach. The asymptotic properties of the proposed estimates are developed through nonparametric score functions. It is presented that nonparametric conditional score functions possess properties similar to those of parametric conditional score functions. This observation leads to simplification of the asymptotic results. The nonparametric MLE of the joint distribution of truncation and censoring variables is also derived. This nonparametric estimate together with the product–limit estimate are used to generalize Efron’s “obvious method” of bootstrapping, for right-censored data, to data that are both left truncated and right censored.