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.

中文翻译：

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Cox偏似然的渐近行为及其在变量选择中的应用

对于 Cox 模型变量选择过程的理论性质，我们研究了 Cox 模型的偏似然渐近行为。我们发现部分似然的行为与普通似然不同，普通似然的样本平均值通常趋于其期望值，即概率上的有限数。在一些温和的条件下，我们证明偏似然的样本平均值以概率的对数的比例趋于无穷大。我们应用偏似然的渐近结果来研究惩罚偏似然的调整参数选择。我们发现，Fan 和 Li (2002) 中提出的广义交叉验证 (GCV) 调整参数的惩罚偏似然具有模型选择一致性属性，尽管 GCV、AIC 和 Cp 在线性回归的情况下是等效的型号，选型不一致。我们通过蒙特卡罗模拟和数据示例进行的实证研究证实了我们的理论发现。