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.'
author:
- 'Xiaoqi Zhang[^1]'
- 'Xiaobing Zhao[^2]'
- 'Yanqiao Zheng[^3]'
title: A novel approach to estimate the Cox model with temporal covariates and application to medical cost data
---

[***Keywords—*** Cox regression; longitudinal process; joint model; maximum full likelihood; adaptive LASSO; Gaussian process; semi-martingale]{}

Introduction
============

In the proportional hazards model ([@cox1972regression; @andersen1982cox]), the hazard function of the event time $T$ takes the form

$$\label{cox-proportional-hazard}
\lambda\left(t\mid Z\right)=\lambda_{0}\left(t\right)\exp\left(b_{0}^{T}Z\right)$$

where $\lambda\left(t\mid Z\right)$ is the conditional hazard function of $T$ given the $p\times1$ covariate vector $Z$, $\lambda_{0}\left(t\right)$ is an unspecified baseline hazard function and $b_{0}$ is a $p\times1$ vector of unknown regression coefficients. Although in the original paper, the covariate $Z$ is viewed as random vectors independent from time $t$, the model can be easily extended to the case where $Z$ is time-dependent and $Z\left(t\right)$ is assumed to be given through an unknown stochastic process.

The main-stream procedure used to estimate the 