In this paper, we consider maximum likelihood estimation of the proportional parameter in a proportional hazard rate (PHR) model based on single and multiply censored order statistics, and progressively Type-II censored order statistics from lifetimes that follow the PHR model. The expectation-maximization (EM) algorithm is proposed for computing the maximum likelihood estimate (MLE) by utilizing the relationship between order statistics and lifetimes of components in a system. The existence and uniqueness of the MLE under different data structures are discussed. We also propose a reliable initial value for the iterative algorithm based on approximating the likelihood function using Taylor series expansion. Monte Carlo simulation studies are used to study the performance of the proposed EM algorithms and the proposed initial value. The proposed EM approach is shown to be a good alternative to the Newton-Raphson method in terms of computational efficient and robustness to the initial value. We also show that the proposed initial value works well with the EM algorithm in obtaining the MLE.

在本文中，我们考虑了基于单次和多次删失阶次统计量以及遵循PHR模型的生命周期逐步进行的II型删失阶次统计量的比例风险率（PHR）模型中比例参数的最大似然估计。提出了期望最大化算法，该算法通过利用顺序统计量与系统组件寿命之间的关系来计算最大似然估计（MLE）。讨论了不同数据结构下MLE的存在性和唯一性。我们还基于泰勒级数展开近似似然函数，为迭代算法提出了可靠的初始值。蒙特卡罗模拟研究用于研究所提出的EM算法的性能和所提出的初始值。就计算效率和对初始值的鲁棒性而言，所提出的EM方法被证明是Newton-Raphson方法的良好替代方案。我们还表明，提出的初始值与EM算法在获得MLE方面效果很好。