Classical simultaneous confidence bands for survival functions (i.e., Hall–Wellner, equal precision, and empirical likelihood bands) are derived from transformations of the asymptotic Brownian nature of the Nelson–Aalen or Kaplan–Meier estimators. Due to the properties of Brownian motion, a theoretical derivation of the highest confidence density region cannot be obtained in closed form. Instead, we provide confidence bands derived from a related optimization problem with local time processes. These bands can be applied to the one-sample problem regarding both cumulative hazard and survival functions. In addition, we present a solution to the two-sample problem for testing differences in cumulative hazard functions. The finite sample performance of the proposed method is assessed by Monte Carlo simulation studies. The proposed bands are applied to clinical trial data to assess survival times for primary biliary cirrhosis patients treated with D-penicillamine.

生存函数的经典同时置信带（即，Hall-Wellner、等精度和经验似然带）源自 Nelson-Aalen 或 Kaplan-Meier 估计量的渐近布朗性质的变换。由于布朗运动的特性，最高置信密度区域的理论推导无法以封闭形式获得。相反，我们提供从本地时间过程的相关优化问题派生的置信带。这些波段可以应用于关于累积风险和生存函数的单样本问题。此外，我们提出了一个解决两个样本问题的方法，用于测试累积风险函数的差异。所提出方法的有限样本性能通过蒙特卡罗模拟研究进行评估。