A semiparametric regression under biased sampling and random censoring, a local pseudo-likelihood approach

Methodologies developed for left-truncated right-censored failure time data can mostly be categorized according to the assumption imposed on the truncation distribution, i.e., being completely unknown or completely known. While the former approach enjoys robustness, the latter is more efficient when the assumed form of the truncation distribution can be supported by the data. Motivated by data from an HIV/AIDS study, we consider the middle ground and develop methodologies for estimation of a regression function in a semiparametric setting where the truncation distribution is parametrically specified while the failure time, censoring, and covariate distribution are left completely unknown. We devise an estimator for the regression function based on a local pseudo-likelihood approach that properly accounts for the bias induced on the response variable and covariate(s) by the sampling design. One important spin-off from these results is that they yield the adjustment for length-biased sampling and right-censoring; the so-called stationary case. We study the small and large sample behaviour of our estimators. The proposed method is then applied to analyze a set of HIV/AIDS data.