Estimation and variable selection for semiparametric transformation models with length-biased survival data

In this study, we investigate estimation and variable selection for semiparametric transformation models with length-biased survival data-a special case of left truncation commonly encountered in the social sciences and cancer prevention trials. To correct for sampling bias, conventional methods such as conditional likelihood, martingale estimating equations, and composite likelihood have been proposed. However, these methods may be less efficient due to their reliance on only partial information from the full likelihood. In contrast, we adopt a full-likelihood approach under the semiparametric transformation model and propose a unified and more efficient nonparametric maximum likelihood estimator (NPMLE). To perform variable selection, we incorporate an adaptive least absolute shrinkage and selection operator (ALASSO) penalty into the full likelihood. We show that when the NPMLE is used as the initial value, the resulting one-step ALASSO estimator-offering a simplified version of the Newton-Raphson method-achieves oracle properties. Theoretical properties of the proposed methods are established using empirical process techniques. The performance of the methods is evaluated through simulation studies and illustrated with a real data application.