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 ,
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