A nonparametric model for product-limit estimation under right censoring and left truncation

The object of this paper is the generalization of the product-limit estimator (PLE) to the case of survival data subject to both right censoring and left truncation. From a practical point of view the problem admits a straightforward solution: The usual Kaplan-Meier formula may be adapted by simply eliminating from the risk set to be assigned to any given point t on the time axis, all items entering observation later than t. The model which is proposed is a natural generalization of the well known random censorship model, giving a precise mathematical meaning to the intuitive notion of a noninformative left truncation mechanism. In the present analysis of the asymptotic behaviour of the PLE under this model, the ‘classical’ approach via empirical processes of i.i.d. random variables is used. The main results derived in this way are strong uniform consistency, and asymptotic normality of the generalized PLE on any compact interval in which the probability of being at risk of failure is bounded away from zero. The proofs are given without relying on continuity of any of the underlying distribution functions.