The Annals of Statistics

Optimal estimation for the functional Cox model

Simeng Qu, Jane-Ling Wang, and Xiao Wang

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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 finite-dimensional 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 $L_{2}$-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.

Article information

Ann. Statist., Volume 44, Number 4 (2016), 1708-1738.

Received: January 2015
Revised: January 2016
First available in Project Euclid: 7 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C20: Minimax procedures 62G05: Estimation 62N01: Censored data models 62N02: Estimation

Cox models functional data minimax rate of convergence partial likelihood right-censored data


Qu, Simeng; Wang, Jane-Ling; Wang, Xiao. Optimal estimation for the functional Cox model. Ann. Statist. 44 (2016), no. 4, 1708--1738. doi:10.1214/16-AOS1441.

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Supplemental materials

  • Supplement to “Optimal estimation for the functional Cox model”. Due to space constraint, the derivation of $\mathit{GCV}(\lambda)$ and proofs of lemmas are relegated to the supplementary file [Qu, Wang and Wang (2016)].