Estimation of Product Reliability for Weibull Distribution under Ranked Set Sampling
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 distri-bution.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 likeli-hood 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 maxi-mum likelihood estimator of the product reliability using ranked set sampling is given.Secondly,the Fisher infor-mation 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 asymptot-ic 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 advan-tage 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 supe-rior 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.
product reliabilityranked set samplingWeibull distributionmaximum likelihood estimatorFisher information matrix
万方数据
维普
