Abstract: The ranked set sampling method is a method for improving sampling efficiency, and it is appropriate for a situation in which the visual ranking of sampling units is easy but the quantification of units is difficult. Although the ranked set sampling protocol was first introduced to solve agriculture problems, it has been applied to various fields since then, including reliable engineering, clinical medicine, ecological environment, etc. Product reliability is an important metric for describing product lifetime. In recent years, some scholars have studied the estimation problem of product reliability under ranked set sampling. Through the comparison of estimation efficiency, it has been proven that the ranked set sampling method is superior to the simple random sampling method. However, most of the product lifetimes studied by these scholars follow an exponential distribution. Weibull distribution plays a very important role in reliability testing, which can describe the lifetime of components, equipment, and other products. When the population distribution is known, the maximum likelihood estimation method is an important method for seeking point estimates and has a wide range of applications.
In order to improve the estimation efficiency of the product reliability for Weibull distribution, this paper studies the maximum likelihood estimation problem of the product reliability using ranked set sampling. Firstly, the maximum likelihood estimators of the shape parameter and scale parameter of Weibull distribution under ranked set sampling are analyzed. On the basis of the invariance of maximum likelihood estimation, the maximum likelihood estimator of the product reliability using ranked set sampling is given. Secondly, the Fisher information matrix of the shape parameter and scale parameter of Weibull distribution under ranked set sampling are computed. The proposed maximum likelihood estimator is shown to have asymptotic normality, and the asymptotic variance of the new estimator is calculated using the Fisher information matrix of the shape parameter and scale parameter. Thirdly, according to the asymptotic variances of the two maximum likelihood estimators under ranked set sampling and simple random sampling, their asymptotic relative efficiencies are analyzed. In order to study the efficiencies of the new estimator and the corresponding estimator under simple random sampling in small samples, a simulation study is performed. Using simulated biases and mean square errors, this paper calculates simulated relative efficiencies of the two maximum likelihood estimators under ranked set sampling and simple random sampling. Finally, the two maximum likelihood estimators under ranked set sampling and simple random sampling are applied to the strength data of single carbon fibers.
The study results of asymptotic relative efficiencies and simulated relative efficiencies show that the proposed estimator is superior to the corresponding estimator based on simple random sampling, and the estimation advantage of the new estimator based on ranked set sampling increases as the set size increases. The actual analysis results of the strength data of single carbon fibers show that the new estimator under ranked set sampling is superior to the corresponding estimator under simple random sampling, which verify the efficiency of the ranked set sampling method.
This paper studies the estimation problem of the product reliability for complete data using ranked set sampling. However, in reliability or survival analysis, the lifetime of interest can be often partially observed. For instance, in medical studies, a subject may die from a cause not related to the study or be lost for a follow-up. In such a case, the lifetime of that subject is truncated. Therefore, in the forthcoming studies, we will consider the estimation problem of the product reliability for truncated data based on ranked set sampling.

Key words: product reliability, ranked set sampling, Weibull distribution, maximum likelihood estimator, Fisher information matrix

摘要： 排序集抽样方法用于样本单元排序容易但测量困难的情形。产品可靠度是描述产品可靠性的重要度量指标,Weibull分布在可靠性试验中占有非常重要的地位。为了提高Weibull分布产品可靠度的估计效率,本文采用排序集抽样方法,研究了产品可靠度的极大似然估计,计算了Weibull分布形状参数和尺度参数的Fisher信息阵,证明了极大似然估计量的渐近正态性,并给出了新估计量的渐近方差。渐近相对效率和模拟相对效率的计算结果表明:排序集抽样下极大似然估计量的估计效率一致高于简单随机抽样下极大似然估计量。最后,单碳纤维强度数据的实际分析结果验证了排序集抽样方法的高效率性。

关键词: 产品可靠度, 排序集抽样, Weibull分布, 极大似然估计, Fisher信息阵