  We propose a novel approach to estimate the Cox model with temporal
covariates. Our new approach treats the temporal covariates as arising from a
longitudinal process which is modeled jointly with the event time. Different
from the literature, the longitudinal process in our model is specified as a
bounded variational process and determined by a family of Initial Value
Problems associated with an Ordinary Differential Equation. Our specification
has the advantage that only the observation of the temporal covariates at the
time to event and the time to event itself are required to fit the model, while
it is fine but not necessary to have more longitudinal observations. This fact
makes our approach very useful for many medical outcome datasets, like the New
York State Statewide Planning and Research Cooperative System and the National
Inpatient Sample, where it is important to find the hazard rate of being
discharged given the accumulative cost but only the total cost at the discharge
time is available due to the protection of patient information. Our estimation
procedure is based on maximizing the full information likelihood function. The
resulting estimators are shown to be consistent and asymptotically normally
distributed. Variable selection techniques, like Adaptive LASSO, can be easily
modified and incorporated into our estimation procedure. The oracle property is
verified for the resulting estimator of the regression coefficients.
Simulations and a real example illustrate the practical utility of the proposed
model. Finally, a couple of potential extensions of our approach are discussed.
