Addressing the End-Cut Preference Problem in Survival Trees with a Smooth Sigmoid Surrogate

The end-cut preference (ECP), the tendency of greedy tree algorithms to select split points near the extremes of a predictor’s range, can produce unstable and highly imbalanced partitions. For survival trees based on the maximized log-rank statistic, we prove that, under the null and an expanding admissible cutpoint range, every maximizing cutpoint leaves any fixed interior quantile interval with probability tending to one. The proof compares the tight interior maximum of a studentized empirical bridge with its diverging maximum over cutpoints approaching an endpoint. An exact variance decomposition of a quadratic Taylor polynomial also identifies a non-negative hard-split term with boundary scale ${(1-2c)}^2{\Lambda }_n/\left\{nc(1-c)\right\}$. For the smooth sigmoid surrogate (SSS), the corresponding Cox-score term is uniformly $O_p(a/n)$, where a is the sigmoid shape parameter. This bound has no endpoint factor. Numerical experiments across censoring levels, signal strengths, split restrictions, and recursive splitting show that SSS generally selects fewer end cuts and yields lower cutpoint RMSE under the alternatives studied, with the clearest gains for weak signals, while retaining the interpretation of ordinary hard splits.