There remain theoretical gaps in deep neural network estimators for the nonparametric Cox proportional hazards model. In particular, it is unclear how gradient-based optimization error propagates to population risk under partial likelihood, how pointwise bias can be controlled to permit valid inference, and how ensemble-based uncertainty quantification behaves under realistic variance decay regimes. We develop an asymptotic distribution theory for deep Cox estimators that addresses these issues. First, we establish nonasymptotic oracle inequalities for general trained networks that link in-sample optimization error to population risk without requiring the exact empirical risk optimizer. We then construct a structured neural parameterization that achieves infinity-norm approximation rates compatible with the oracle bound, yielding control of the pointwise bias. Under these conditions and using the Hajek--Hoeffding projection, we prove pointwise and multivariate asymptotic normality for subsampled ensemble estimators. We derive a range of subsample sizes that balances bias correction with the requirement that the Hajek--Hoeffding projection remain dominant. This range accommodates decay conditions on the single-overlap covariance, which measures how strongly a single shared observation influences the estimator, and is weaker than those imposed in the subsampling literature. An infinitesimal jackknife representation provides analytic covariance estimation and valid Wald-type inference for relative risk contrasts such as log-hazard ratios. Finally, we illustrate the finite-sample implications of the theory through simulations and a real data application.

翻译：深度神经网络估计器在非参数Cox比例风险模型中仍存在理论空白。具体而言：基于梯度的优化误差如何在偏似然框架下传播至总体风险、如何控制逐点偏差以实现有效推断，以及基于集成的置信区间量化在现实方差衰减机制下的行为特征等问题尚未明确。针对上述问题，我们建立了深度Cox估计器的渐近分布理论。首先，我们构建了通用训练网络的非渐近预言不等式，在无需精确经验风险优化器的条件下建立了样本内优化误差与总体风险的联系。继而构造了与预言界相容且具有无穷范数逼近速率的结构化神经参数化方法，实现了对逐点偏差的控制。基于上述条件并借助Hajek--Hoeffding投影，我们证明了子采样集成估计器的逐点及多元渐近正态性。推导了能在偏差校正与保持Hajek--Hoeffding投影主导地位之间取得平衡的子采样规模范围，该范围可容纳衡量单次共同观测对估计器影响强度的单重叠协方差的衰减条件，其约束强度弱于子采样文献中的常见假设。通过无穷小刀切表示法实现了解析协方差估计，并为对数风险比等相对风险对比量提供了有效的Wald型推断。最后，通过仿真实验与真实数据应用验证了理论在有限样本场景下的有效性。