Asymptotic behavior of Cox's partial likelihood and its application to variable selection

For theoretical properties of variable selection procedures for Cox's model, we study the asymptotic behavior of partial likelihood for the Cox model. We find that the partial likelihood does not behave like an ordinary likelihood, whose sample average typically tends to its expected value, a finite number, in probability. Under some mild conditions, we prove that the sample average of partial likelihood tends to infinity at the rate of the logarithm of the sample size, in probability. We apply the asymptotic results on the partial likelihood to study tuning parameter selection for penalized partial likelihood. We find that the penalized partial likelihood with the generalized cross-validation (GCV) tuning parameter proposed in Fan and Li (2002) enjoys the model selection consistency property, despite the fact that GCV, AIC and Cp, equivalent in the context of linear regression models, are not model selection consistent. Our empirical studies via Monte Carlo simulation and a data example confirm our theoretical findings.