Estimating high‐dimensional additive Cox model with time‐dependent covariate processes

Abstract

This paper is concerned with the estimation in the additive Cox model with time-dependent covariates when the number of additive components p is greater than the sample size n. By combining spline representation and the group lasso KEYWORDS

Estimation and inference for high-dimensional nonparametric additive instrumental-variables regression

“…Assumption 2 is rather standard in the literature of high dimensional additive modeling [23,11]. Assumption 3 is often called group-wise restricted eigenvalue assumption [28,30,29]. This is a natural extension of the restricted eigenvalue assumption for the usual lasso and Dantzig selector problems [4].…”

Semiparametric Model Averaging Prediction for Lifetime Data via Hazards Regression

“…To be specific, one can obtain the regularized estimates ̂ from the penalized log-partial likelihood, which is given by ̂ = arg max L a ( ) − a J( ) where L a is given in Equation (2) and J is a specified penalty function such as group LASSO. Lv et al (2018) and Yuan et al (2018) offered some theoretical justifications for such penalized log-partial likelihood with specified degrees of freedom of covariates. The sparse solution directly leads to the prediction ̂ a .…”

“…Some recent authors also considered other ad hoc specification for covariates’ degrees of freedom, for example, Lv et al. (2018) and Yuan et al. (2018).…”

“…It is computationally prohibitive to simultaneously tune all the components using AIC or BIC. Some recent authors also considered other ad hoc specification for covariates' degrees of freedom, for example, Lv et al (2018) and Yuan et al (2018). We stress that if the goal is to provide an accurate prediction for the response, using a single model, despite its complexity, may still suffer from model mis-specification and overlook information beyond the assumed dependence structure in this model.…”

“…Allowing the covariate effects to be unspecified functions, the nonparametric estimation in an NPH model generally can be achieved by one of the two well‐known fitting methods: the kernel regression method (see Chen & Zhou, 2007; Chen et al., 2010; Fan et al., 1997; Tibshirani & Hastie, 1987), and the splines basis approximation method (see Cao et al., 2016; Gray, 1992; Kooperberg et al., 1995; Lv et al., 2018; Sleeper & Harrington, 1990; Yang et al., 2018; Yuan et al., 2018). Both methods receive strong theoretical support and are widely adopted in various applications.…”