In this paper, we study theoretical properties of the non-parametric Cox proportional hazards model in a high dimensional non-asymptotic setting. We establish the finite sample oracle $l_{2}$ bounds for a general class of group penalties that allow possible hierarchical and overlapping structures. We approximate the log partial likelihood with a quadratic functional and use truncation arguments to reduce the error. Unlike the existing literature, we exemplify differences between bounded and possibly unbounded non-parametric covariate effects. In particular, we show that bounded effects can lead to prediction bounds similar to the simple linear models, whereas unbounded effects can lead to larger prediction bounds. In both situations we do not assume that the true parameter is necessarily sparse. Lastly, we present new theoretical results for hierarchical and smoothed estimation in the non-parametric Cox model. We provide two examples of the proposed general framework: a Cox model with interactions and an ANOVA type Cox model.
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机译：在本文中，我们研究了在高维非渐近环境下非参数Cox比例风险模型的理论特性。我们为组罚的一般类别建立了有限的示例oracle $ l {{2} $界限，该类别允许可能的层次结构和重叠结构。我们用二次函数近似对数偏似性，并使用截断参数来减少误差。与现有文献不同，我们举例说明有界和可能无界的非参数协变量效应之间的差异。特别是，我们证明了有界效应可以导致类似于简单线性模型的预测界限，而无界效应可以导致更大的预测界限。在这两种情况下，我们都不认为true参数必然是稀疏的。最后，我们为非参数Cox模型中的分层和平滑估计提供了新的理论结果。我们提供了所建议的通用框架的两个示例：具有交互作用的Cox模型和ANOVA类型的Cox模型。
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