Abstract In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.

中文翻译：

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指数分布应力-强度可靠性参数的最小风险序列点估计

摘要 在本文中，使用纯和两阶段顺序程序，考虑了应力-强度模型下可靠性参数（R）的最小风险点估计问题，在损失函数为平方误差加抽样成本的情况下，当随机应力 (X) 和随机强度 (Y) 是独立的，并且都具有不同尺度参数的指数分布。在两阶段顺序程序下提供了总样本量的精确分布以及应力-强度模型下可靠性参数的最大似然估计量的期望值和均方误差的显式公式。使用大数定律和蒙特卡洛积分，近似了纯顺序程序下停止规则的精确分布。而且，结果表明，两个提出的顺序程序都是有限的，并且对于特殊情况，停止时间的确切分布在初始样本大小处具有退化分布。在仿真的帮助下研究了所提出方法的性能。最后，使用真实的数据集，清楚地说明了程序。