Improving efficiency using the Rao-Blackwell theorem in corrected and conditional score estimation methods for joint models

Longitudinal covariates in survival models are generally analyzed using random effects models. By framing the estimation of these survival models as a functional measurement error problem, semiparametric approaches such as the conditional score or the corrected score can be applied to find consistent estimators for survival model parameters without distributional assumptions on the random effects. However, in order to satisfy the standard assumptions of a survival model, the semiparametric methods in the literature only use covariate data before each event time. This suggests that these methods may make inefficient use of the longitudinal data. We propose an extension of these approaches that follows a generalization of Rao-Blackwell theorem. A Monte Carlo error augmentation procedure is developed to utilize the entirety of longitudinal information available. The efficiency improvement of the proposed semiparametric approach is confirmed theoretically and demonstrated in a simulation study. A real data set is analyzed as an illustration of a practical application