Doubly censored prevalent cohort survival data

When survival data are collected as part of a prevalent cohort study, the recruited cases have already experienced their initiating event. When these initiating times are generated from a Stationary Poisson Process, these survival data are termed "length-biased". Our interest is to estimate the Survival function, from the initiating event time, which is often known only with uncertainty, to the failure or censoring time. We derive the likelihood function and propose two methods to estimate the survival function incorporating the possible uncertainty. We verify the methods by the means of simulation. We also illustrate the methods using a real set of data on survival with dementia collected as part of the Canadian Study of Health and Aging (CSHA)