The relationship between a time-dependent covariate and survival times is usually evaluated via the Cox model. Time-dependent covariates are generally available as longitudinal data collected regularly during the course of the study. A frequent problem, however, is the occurence of missing covariate data. A recent approach to estimation in the Cox model in this case jointly models survival and the longitudinal covariate. However, theoretical justification of this approach is still lacking. In this paper we prove existence and consistency of the maximum likelihood estimators in a joint model. The asymptotic distribution of the estimators is given along with a consistent estimator of the asymptotic variance.
"Asymptotic theory for the Cox model with missing time-dependent covariate." Ann. Statist. 34 (2) 903 - 924, April 2006. https://doi.org/10.1214/009053606000000038