Some asymptotic results of survival tree and forest models

This paper develops a theoretical framework and asymptotic results for survival tree and forest models under right censoring. We first investigate the method from the aspect of splitting rules, where the survival curves of the two potential child nodes are calculated and compared. We show that existing approaches lead to a potentially biased estimation of the within-node survival and cause non-optimal selection of the splitting rules. This bias is due to the censoring distribution and the non i.i.d. sample structure within each node. Based on this observation, we develop an adaptive concentration bound result for both tree and forest versions of the survival tree models. The result quantifies the variance component for survival forest models. Furthermore, we show with three specific examples how these concentration bounds, combined with properly designed splitting rules, yield consistency results. The three examples are: 1) a finite dimensional setting with random splitting rules; 2) an infinite dimensional case with marginal signal checking; and 3) an infinite dimensional setting with principled Cox screening splitting rule. The development of these results serves as a general framework for showing the consistency of tree- and forest-based survival models.