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.

翻译：形状限制推断在生存数据的多种应用中已展现出实证优势。然而，部分研究未能提供严格的理论论证及具备理论保证的易用方差估计量。受Deng等人（2023）研究的启发，本文深入探讨了针对右删失数据的可加且形状限制的部分线性Cox模型，其中每个可加成分均满足特定形状约束，包括单调递增/递减及凸性/凹性。我们系统研究了所有基础参数的形状限制最大部分似然估计量（SMPLE）的相合性与收敛速率。进一步建立了线性协变量偏移情形下SMPLE的渐近正态性与半参数有效性。为估计渐近方差，我们提出了一种创新的数据分割方差估计方法，该方法具有卓越的通用性与广泛适用性。模拟结果与鹿特丹乳腺癌数据集分析表明：当Cox模型正确时，SMPLE与Cox模型下的最大似然估计量性能相当；而当Cox模型被违反或风险函数非光滑时，SMPLE的表现优于后者及Huang（1999）的方法。同时，所提出的方差估计方法通常能为SMPLE及其竞争方法提供可靠的区间估计。