A class of variable selection procedures for parametric models via
nonconcave penalized likelihood was proposed in Fan and Li (2001a). It has been
shown there that the resulting procedures perform as well as if the subset of
significant variables were known in advance. Such a property is called an
oracle property. The proposed procedures were illustrated in the context of
linear regression, robust linear regression and generalized linear models. In
this paper, the nonconcave penalized likelihood approach is extended further to
the Cox proportional hazards model and the Cox proportional hazards frailty
model, two commonly used semi-parametric models in survival analysis. As a
result, new variable selection procedures for these two commonly-used models
are proposed. It is demonstrated how the rates of convergence depend on the
regularization parameter in the penalty function. Further, with a proper choice
of the regularization parameter and the penalty function, the proposed
estimators possess an oracle property. Standard error formulae are derived and
their accuracies are empirically tested. Simulation studies show that the
proposed procedures are more stable in prediction and more effective in
computation than the best subset variable selection, and they reduce model
complexity as effectively as the best subset variable selection. Compared with
the LASSO, which is the penalized likelihood method with the $L_1$ -penalty,
proposed by Tibshirani, the newly proposed approaches have better theoretic
properties and finite sample performance.