  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.
