An optimal completion of the product limit estimator

It is well known that the product limit estimator is undefined beyond the largest observation if it is censored. Some completion methods are suggested in the literature (see e.g. [Efron, 1967. The two sample problem with censored data. Proceedings of the 5th Berkeley Symposium] and [Gill, 1980. Censoring and stochastic integrals. Mathematical Centre Tract No. 124, Mathematisch Centrum, Amsterdam]). In this note, we propose a completion method that is optimal in the sense that the expected value of the integrated squared error loss function is minimized. This method yields an estimator that falls between the above two extremes and possesses the same large sample properties. New bounds for the biases are also derived for the above-mentioned cases.