Sampling in survival analysis and estimation with unknown selection bias and censoring

In a population of individuals, where the random variable (r.v.) sigma denotes the birth time and X the lifetime, we consider that an individual can be observed only if its life-line L(sigma, X) = {(sigma + y, y), 0 <= y <= X} intersects a general Borel set S in R x R+. Denoting by sigma(S) and X-S the birth time and lifetime for the observed individuals, we point out that the distribution function (d.f.) F-S of the r.v. X-S suffers from a selection bias in the sense that F-S = integral wdF/mu(S), where w and mu(S) depend only on the distribution of sigma and F is the d.f. of X. Considering in addition that the r.v. X-S is randomly right-censored, as soon as the individual is selected, we construct a product-limit estimator (F) over cap (S) for the d.f. F-S and a nonparametric estimator (w) over cap for the weighting function w. We investigate the behavior of these estimators through a simulation study.