The Generalized Lower Confidence Limits Of System Reliability Under Weibull Distribution And Gamma Distribution

Weibull distribution and Gamma distribution are two important distributions in statistics,which are widely used in reliability and other fields.At present,scholars at home and abroad mainly focus on parameter estimation,there are relatively few studies on the reliability of series(parallel)systems under the Weibull and Gamma distributions.At the same time,due to the continuous innovation of Science and technology,the structure of the system becomes more and more complex,and the reliability of the system becomes more and more prominent,therefore,how to judge and study the reliability of the system quickly,effectively and accurately,estimate the actual performance of the system correctly,and reduce the risk of the system has become a common concern of experts and scholars all over the world.And in some practical problems,for the system reliability research,people are more concerned about its lower limit value,therefore,based on the life data of system components,in this paper,the generalized lower confidence limits of the reliability of series(parallel)systems are discussed when the life distributions of components are Weibull distribution and Gamma distribution respectively.First of all,under the condition of the Weibull distribution of component life,under the type truncated life test,the shape parameters are the same,the scale parameters are different and the shape parameters are different,the shape parameters and the scale parameters are maximum likelihood under different conditions,and then the reliability maximum likelihood of the series(parallel)system are obtained according to the maximum likelihood invariance.Thirdly,based on Weerahandi’s generalized pivot quantity method,the generalized pivot quantity and the generalized lower confidence limit of the series(parallel)system reliability under the above two parameters are given respectively,the coverage of the generalized lower confidence limit of the series system reliability is slightly higher than the nominal coverage,so it is corrected by Fisher-z transform.The generalized lower confidence limits for the reliability of series systems with the same shape parameters and different scale parameters need to be arithmetic mean by taking the values of the lower confidence limits before and after the correction,the simulation results show that the re-correction effect is significant.For the parallel mode,the coverage of the generalized lower confidence limit of the reliability of the parallel system under both parameters is close to the nominal coverage,and no correction is needed.Then,in the case that the lifetime of a component obeys the Gamma distribution,under the life test of a complete sample,the shape parameters are the same,the scale parameters are different and the shape parameters are different,the shape parameters and the scale parameters are maximum likelihood under different conditions,and then the reliability maximum likelihood of the series(parallel)system are obtained according to the maximum likelihood invariance.Thirdly,based on Cornish-Fisher expansion,cumulative distribution function theorem and Weerahandi’s generalized pivot method,the generalized lower confidence limits for the reliability of series(parallel)systems are obtained,in order to solve the problem that the lower bound of the generalized confidence of the system reliability is slightly lower in series mode,the method is modified by Fisher-z transform.For parallel mode,no modification is required under both conditions.