Many authors have considered the problem of estimating a distribution function when the observed data is subject to random truncation. A prominent role is played by the product limit estimator, which is the analogue of the Kaplan-Meier estimator of a distribution function under random censoring. Wang and Jewell (1985) and Woodroofe (1985) independently proved consistency results for this product limit estimator and showed weak convergence to a Gaussian process. Both papers left open the exact form of the covariance structure of the limiting process. Here we provide a precise description of the asymptotic behavior of the product limit estimator, including a simple explicit form of the asymptotic covariance structure, which also turns out to be the analogue of the covariance structure of the Kaplan-Meier estimator. Some applications are briefly discussed.
"Asymptotic Properties of the Product Limit Estimate Under Random Truncation." Ann. Statist. 14 (4) 1597 - 1605, December, 1986. https://doi.org/10.1214/aos/1176350180