It is known that length-biased survival data commonly arise in cross-sectional surveys and prevalent cohort studies on disease duration. It is also known that in survival analysis the risk pattern of a disease, mortality or other events in time are usually explored via the survival or hazard function. In the case of investigating the risk of failure a careful study of any possible change in the hazard function of failure is more helpful since it provides a refined insight into the structural changes of the risk pattern of a time-related event.
Since now the estimation of a change-point in the hazard function has been studied in parametric, nonparametric or semi-parametric setting considering only representative samples from the population of interest which limited to incident cases. However when observations form a biased sample from the target population the usage of methods that account for left truncation distribution \(H\) in addition to informative censoring is much more helpful.
Following the above ideas the authors managed to investigate nonparametric methods for estimating a possible change-point (location and size) of an otherwise smooth hazard function under biased sampling (from the target population) in the case that the observations are subject to informative censoring. To this direction and by considering two scenarios regarding the left-truncation distribution \(H\) (known and unknown) they managed to provide two types of estimators regarding the estimation of the maximum modulus on the convolution \(K_*(\cdot/h_n)/h_n\) with the hazard function, where \(K_*\) is an appropriate odd function and \((h_n)_n\) a sequence of bandwidth tending to zero. Although the estimators in the first case which incorporate the available information of \(H\) were found more efficient compared to those in the second case where \(H\) is unknown, the latter case provides more robust estimators, in the form of the truncation distribution. However in both cases the change-point estimators can achieve the rate \(O_p(1/n)\).
Furthermore the authors managed to study the asymptotic distributions in both cases, to devise interval-estimators for the location and size of the change paving the way towards making statistical inference about whether or not a change-point exists, to study the finite sample behavior of the estimators through several simulated examples as well as to illustrate the two proposed methods via the analysis of a set of survival data collected on elderly Canadian citizens (aged \(65^+\)) suffering from dementia.