  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.
