In this paper, we consider two well-known parametric long-term survival models, namely, the Bernoulli cure rate model and the promotion time (or Poisson) cure rate model. Assuming the long-term survival probability to depend on a set of risk factors, the main contribution is in the development of the stochastic expectation maximization (SEM) algorithm to determine the maximum likelihood estimates of the model parameters. We carry out a detailed simulation study to demonstrate the performance of the proposed SEM algorithm. For this purpose, we assume the lifetimes due to each competing cause to follow a two-parameter generalized exponential distribution. We also compare the results obtained from the SEM algorithm with those obtained from the well-known expectation maximization (EM) algorithm. Furthermore, we investigate a simplified estimation procedure for both SEM and EM algorithms that allow the objective function to be maximized to split into simpler functions with lower dimensions with respect to model parameters. Moreover, we present examples where the EM algorithm fails to converge but the SEM algorithm still works. For illustrative purposes, we analyze a breast cancer survival data. Finally, we use a graphical method to assess the goodness-of-fit of the model with generalized exponential lifetimes.

在本文中，我们考虑了两个著名的参数化长期生存模型，即伯努利治愈率模型和促进时间（或泊松）治愈率模型。假设长期生存概率取决于一组风险因素，则主要贡献在于开发了随机期望最大化（SEM）算法，以确定模型参数的最大似然估计。我们进行了详细的仿真研究，以证明所提出的SEM算法的性能。为此，我们假设由于每个竞争原因而导致的寿命遵循两参数的广义指数分布。我们还将从SEM算法获得的结果与从众所周知的期望最大化（EM）算法获得的结果进行比较。此外，我们研究了针对SEM和EM算法的简化估计程序，该程序允许将目标函数最大化，从而相对于模型参数以较小的维度拆分为更简单的函数。此外，我们提供了EM算法无法收敛但SEM算法仍然有效的示例。出于说明目的，我们分析了乳腺癌的生存数据。最后，我们使用图形化方法评估具有广义指数寿命的模型的拟合优度。