Random survival forest and survival trees are popular models in statistics and machine learning. However, there is a lack of general understanding regarding consistency, splitting rules and influence of the censoring mechanism. In this paper, we investigate the statistical properties of existing methods from several interesting perspectives. First, we show that traditional splitting rules with censored outcomes rely on a biased estimation of the within-node failure distribution. To exactly quantify this bias, we develop a concentration bound of the within-node estimation based on non i.i.d. samples and apply it to the entire forest. Second, we analyze the entanglement between the failure and censoring distributions caused by univariate splits, and show that without correcting the bias at an internal node, survival tree and forest models can still enjoy consistency under suitable conditions. In particular, we demonstrate this property under two cases: a finite-dimensional case where the splitting variables and cutting points are chosen randomly, and a high-dimensional case where the covariates are weakly correlated. Our results can also degenerate into an independent covariate setting, which is commonly used in the random forest literature for high-dimensional sparse models. However, it may not be avoidable that the convergence rate depends on the total number of variables in the failure and censoring distributions. Third, we propose a new splitting rule that compares bias-corrected cumulative hazard functions at each internal node. We show that the rate of consistency of this new model depends only on the number of failure variables, which improves from non-bias-corrected versions. We perform simulation studies to confirm that this can substantially benefit the prediction error.

中文翻译：

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生存树和森林模型的一致性：分裂偏差和校正

随机生存森林和生存树是统计学和机器学习中的流行模型。然而，对于审查机制的一致性、分裂规则和影响缺乏普遍的理解。在本文中，我们从几个有趣的角度研究了现有方法的统计特性。首先，我们表明具有删失结果的传统分裂规则依赖于对节点内故障分布的有偏估计。为了准确量化这种偏差，我们基于非 iid 样本开发了节点内估计的浓度界限，并将其应用于整个森林。其次，我们分析了由单变量分裂引起的失败和审查分布之间的纠缠，并表明在没有纠正内部节点的偏差的情况下，生存树和森林模型在合适的条件下仍然可以保持一致性。特别是，我们在两种情况下证明了这一特性：随机选择分裂变量和切割点的有限维情况，以及协变量弱相关的高维情况。我们的结果也可以退化为独立的协变量设置，这通常用于高维稀疏模型的随机森林文献中。然而，收敛速度取决于失效分布和删失分布中变量的总数可能无法避免。第三，我们提出了一个新的分裂规则，该规则比较每个内部节点的偏差校正累积风险函数。我们表明这个新模型的一致性率仅取决于失效变量的数量，这是从非偏差校正版本改进的。我们进行模拟研究以确认这可以大大有益于预测误差。