  We study the Cox models with semiparametric relative risk, which can be
partially linear with one nonparametric component, or multiple additive or
nonadditive nonparametric components. A penalized partial likelihood procedure
is proposed to simultaneously estimate the parameters and select variables for
both the parametric and the nonparametric parts. Two penalties are applied
sequentially. The first penalty, governing the smoothness of the multivariate
nonlinear covariate effect function, provides a smoothing spline ANOVA
framework that is exploited to derive an empirical model selection tool for the
nonparametric part. The second penalty, either the
smoothly-clipped-absolute-deviation (SCAD) penalty or the adaptive LASSO
penalty, achieves variable selection in the parametric part. We show that the
resulting estimator of the parametric part possesses the oracle property, and
that the estimator of the nonparametric part achieves the optimal rate of
convergence. The proposed procedures are shown to work well in simulation
experiments, and then applied to a real data example on sexually transmitted
diseases.
