The lifetime distributions with bathtub-shaped hazard rate functions and censoring scheme have been used widely in life testing and reliability engineering. This paper develops a new approach for estimating parameters of an important two-parameter lifetime data analysis model with bathtub-shaped hazard rate function under the assumption that sample is modified progressively hybrid censored. One of the most frequently used methodologies, maximum likelihood (ML) estimation, is used for estimating unknown parameters. The estimates of unknown parameters are proposed using popular Newton–Raphson algorithm because the estimators cannot be obtained in closed forms. It is well known that the convergence of Newton–Raphson algorithm is affected by an initial point. Therefore, a new Newton–Raphson algorithm with an adaptive initial point within the exact joint confidence region has been suggested to compute the ML estimation. Extensive numerical simulations show that the proposed algorithm converges all the times and it is effective. Finally, one real-world data set from engineering is analysed to illustrate the application of the proposed method.

中文翻译：

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一种改进渐进混合删失下估计双参数浴缸形寿命分布参数的新方法

具有浴缸形危险率函数和删失方案的寿命分布已广泛用于寿命测试和可靠性工程。本文开发了一种新方法，用于估计具有浴缸形风险率函数的重要双参数寿命数据分析模型的参数，该模型假设样本是逐步修改的，混合删失。最常用的方法之一是最大似然 (ML) 估计，用于估计未知参数。未知参数的估计是使用流行的 Newton-Raphson 算法提出的，因为估计量不能以封闭形式获得。众所周知，Newton-Raphson 算法的收敛性受初始点的影响。所以，已经建议使用在精确联合置信区域内具有自适应初始点的新 Newton-Raphson 算法来计算 ML 估计。大量的数值模拟表明，所提出的算法始终收敛并且是有效的。最后，分析了一个来自工程的真实世界数据集，以说明所提出方法的应用。