Optimal estimation for the functional Cox model

Abstract

Functional covariates are common in many medical, biodemographic, and neuroimaging studies. The aim of this paper is to study functional Cox models with right-censored data in the presence of both functional and scalar covariates. We study the asymptotic properties of the maximum partial likelihood estimator and establish the asymptotic normality and efficiency of the estimator of the finitedimensional estimator. Under the framework of reproducing kernel Hilbert space, the estimator of the coefficient function for a functional covariate achieves the minimax optimal rate of convergence under a weighted L2-risk. This optimal rate is determined jointly by the censoring scheme, the reproducing kernel and the covariance kernel of the functional covariates. Implementation of the estimation approach and the selection of the smoothing parameter are discussed in detail. The finite sample performance is illustrated by simulated examples and a real application.J is a penalty function controlling the smoothness of β, and λ is a smoothing parameter that balances the fidelity to the model and the plausibility of β. The choice of the penalty function J(·) is a squared semi-norm associated with H and its norm. In general, H(K) can be decomposed with respect to the penalty J as H = N J + H 1 , where N J is the null space defined asand H 1 is its orthogonal complement in H. Correspondingly, the kernel K can be decomposed as K = K 0 + K 1 , where K 0 and K 1 are kernels for the subspace N J and H 1 respectively. For example, for the Sobolev space, W 2,m = f : [0, 1] → R| f, f , . . . f (m−1) are absolutely continuous, f (m) ∈ L 2 ,

FLCRM: Functional Linear Cox Regression Model

“…However, for the method of Qu et al. (), it fails to estimate and accurately. We guess that it may be caused by not using the traditional Newton–Raphson algorithm to obtain the estimator from penalized cox regression model.…”

“…To examine the effects of r n on the estimation of parameters, we varied r n from 1 to 10. We have also compared our method with the methods proposed in Gellar et al (2015) and Qu et al (2016). We include the estimation results for n = 200 and censoring rate 0.1 in Table 1.…”

“…() and Qu et al. () proposed to maximize penalized partial likelihood functions for model , whereas Lee et al. () developed a Bayesian framework for the same model.…”

“…() combined penalized signal regression with methods developed for mixed effects proportional hazards models under penalized B‐spline framework, and Qu et al. () estimated the model under the reproducing kernel Hilbert space framework.…”

“…At the time of submission, we are aware of three recent articles on the development of various estimation methods for model (1.3). Gellar et al (2015) and Qu et al (2016) proposed to maximize penalized partial likelihood functions for model (1.3), whereas Lee et al (2015) developed a Bayesian framework for the same model. In particular, Gellar et al (2015) combined penalized signal regression with methods developed for mixed effects proportional hazards models under penalized B-spline framework, and Qu et al (2016) estimated the model under the reproducing kernel Hilbert space framework.…”