A central limit theorem for functionals of the Kaplan--Meier estimator

A central limit theorem is given for functionals of the Kaplan--Meier estimator when the censoring distributions are possibly different or discontinuous. In the i.i.d. case only one easily interpretable and simple integrability condition is needed. This condition reduces to the usual condition for the Lindeberg--Lévy theorem when there is no censoring; it is also necessary in certain other situations. The weak convergence of the corresponding processes is also established. The simple proofs and conditions result from the martingale method of Gill (1983), an extension of an identity of Shorack and Wellner (1986) and a delicate treatment of the remainder terms.