  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.
