Optimal Estimation for the Functional Cox Model

TL;DR

This paper develops an optimal estimation method for the functional Cox model with both functional and scalar covariates, establishing asymptotic properties and demonstrating minimax optimal convergence rates.

Contribution

It introduces a new estimation approach for the functional Cox model within a reproducing kernel Hilbert space framework, achieving optimal convergence rates.

Findings

Estimator is asymptotically normal and efficient.

Achieves minimax optimal rate of convergence.

Demonstrates good finite sample performance through simulations and real data.

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 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.