  Shape-restricted inferences have exhibited empirical success in various
applications with survival data. However, certain works fall short in providing
a rigorous theoretical justification and an easy-to-use variance estimator with
theoretical guarantee. Motivated by Deng et al. (2023), this paper delves into
an additive and shape-restricted partially linear Cox model for right-censored
data, where each additive component satisfies a specific shape restriction,
encompassing monotonic increasing/decreasing and convexity/concavity. We
systematically investigate the consistencies and convergence rates of the
shape-restricted maximum partial likelihood estimator (SMPLE) of all the
underlying parameters. We further establish the aymptotic normality and
semiparametric effiency of the SMPLE for the linear covariate shift. To
estimate the asymptotic variance, we propose an innovative data-splitting
variance estimation method that boasts exceptional versatility and broad
applicability. Our simulation results and an analysis of the Rotterdam Breast
Cancer dataset demonstrate that the SMPLE has comparable performance with the
maximum likelihood estimator under the Cox model when the Cox model is correct,
and outperforms the latter and Huang (1999)'s method when the Cox model is
violated or the hazard is nonsmooth. Meanwhile, the proposed variance
estimation method usually leads to reliable interval estimates based on the
SMPLE and its competitors.
