当代的变点生存模型主要关注整个研究人群共享的未知普遍变点。然而，在某些情况下，变点很可能是个体特定的，比如它与端粒长度或绝经年龄相对应的情况。此外，基于最大似然的固定变点参数推断复杂而闻名。最大似然估计器的渐近分布是非标准的，常常使用计算密集的引导技术来恢复其采样分布。本文是在乳腺癌研究的驱动下进行的，该研究假设患者的无病生存时间受到未观测的绝经年龄的调节。由于绝经年龄在患者之间变化，固定变点生存模型可能是不适当的。因此，我们提出了一种具有随机变点的新型比例风险模型。我们开发了一种非参数最大似然估计方法，并设计了一种稳定的期望最大化算法来计算估计值。由于该模型是正则的，我们使用传统的似然理论进行推断，基于欧几里得参数估计值的渐近正态性，并且渐近分布的方差可以通过概要似然方法进行一致估计。模拟研究证明了所提方法的满意的有限样本性能，得到了小偏差和适当的覆盖概率。该方法应用于了乳腺癌研究中。

Contemporary works in change-point survival models mainly focus on an unknown universal change-point shared by the whole study population. However, in some situations, the change-point is plausibly individual-specific, such as when it corresponds to the telomere length or menopausal age. Also, maximum-likelihood-based inference for the fixed change-point parameter is notoriously complicated. The asymptotic distribution of the maximum-likelihood estimator is non-standard, and computationally intensive bootstrap techniques are commonly used to retrieve its sampling distribution. This article is motivated by a breast cancer study, where the disease-free survival time of the patients is postulated to be regulated by the menopausal age, which is unobserved. As menopausal age varies across patients, a fixed change-point survival model may be inadequate. Therefore, we propose a novel proportional hazards model with a random change-point. We develop a nonparametric maximum-likelihood estimation approach and devise a stable expectation-maximization algorithm to compute the estimators. Because the model is regular, we employ conventional likelihood theory for inference based on the asymptotic normality of the Euclidean parameter estimators, and the variance of the asymptotic distribution can be consistently estimated by a profile-likelihood approach. A simulation study demonstrates the satisfactory finite-sample performance of the proposed methods, which yield small bias and proper coverage probabilities. The methods are applied to the motivating breast cancer study.