  Right censored survival data collected on a cohort of prevalent cases with
constant incidence are length-biased, and may be used to estimate the
length-biased (i.e., prevalent-case) survival function. When the incidence rate
is constant, so-called stationarity of the incidence, it is more efficient to
use this structure for unconditional statistical inference than to carry out an
analysis by conditioning on the observed truncation times. It is well known
that, due to the informative censoring for prevalent cohort data, the
Kaplan--Meier estimator is not the unconditional NPMLE of the length-biased
survival function and the asymptotic properties of the NPMLE do not follow from
any known result. We present here a detailed derivation of the asymptotic
properties of the NPMLE of the length-biased survival function from right
censored prevalent cohort survival data with follow-up. In particular, we show
that the NPMLE is uniformly strongly consistent, converges weakly to a Gaussian
process, and is asymptotically efficient. One important spin-off from these
results is that they yield the asymptotic properties of the NPMLE of the
incident-case survival function [see Asgharian, M'Lan and Wolfson J. Amer.
Statist. Assoc. 97 (2002) 201--209], which is often of prime interest in a
prevalent cohort study. Our results generalize those given by Vardi and Zhang
[Ann. Statist. 20 (1992) 1022--1039] under multiplicative censoring, which we
show arises as a degenerate case in a prevalent cohort setting.
