论文标题
高维Cox回归模型的现代理论
A Modern Theory for High-dimensional Cox Regression Models
论文作者
论文摘要
比例危害模型已在许多领域（例如生物医学）进行了广泛使用，以估算和执行统计显着性测试对影响患者生存时间的协变量的影响。大多数软件包使用最大部分样品估计（MPLE）的经典理论来产生推理，例如，R中的CoxPH函数和SAS中的PHREG程序。在本文中，我们研究了参数p的数量P与样品n数量相同的序列中MPLE的渐近行为。主要结果是（i）MPLE的存在是急剧的“相转换”； （ii）经典的理论导致高维度的推断无效。我们表明，MPLE的渐近行为受到新的渐近理论的控制。这些发现通过数值研究进一步证实。我们证明的主要技术工具是凸高斯最低 - 最大定理（CGMT），该定理先前尚未用于部分可能性分析。因此，我们的结果扩大了CGMT的范围，并为使用CGMT检查了MPLE和不可分割的目标函数的存在而开发了新的启示。
The proportional hazards model has been extensively used in many fields such as biomedicine to estimate and perform statistical significance testing on the effects of covariates influencing the survival time of patients. The classical theory of maximum partial-likelihood estimation (MPLE) is used by most software packages to produce inference, e.g., the coxph function in R and the PHREG procedure in SAS. In this paper, we investigate the asymptotic behavior of the MPLE in the regime in which the number of parameters p is of the same order as the number of samples n. The main results are (i) existence of the MPLE undergoes a sharp 'phase transition'; (ii) the classical MPLE theory leads to invalid inference in the high-dimensional regime. We show that the asymptotic behavior of the MPLE is governed by a new asymptotic theory. These findings are further corroborated through numerical studies. The main technical tool in our proofs is the Convex Gaussian Min-max Theorem (CGMT), which has not been previously used in the analysis of partial likelihood. Our results thus extend the scope of CGMT and shed new light on the use of CGMT for examining the existence of MPLE and non-separable objective functions.
