Research on the Ststistical Inference and δ-shock Model in Repairable Systems
|Course||Probability Theory and Mathematical Statistics|
|Keywords||repairable system δ-shock model α-series process Bayesestimation parameter estimation interval estimation|
The repairable problem has been an important part of the reliabilitytheory for a long period. In these decades, many investigators have beenstudied different kinds of reliability systems and got many reliability indexesto judge the property of the systems. Furthermore, they also have got the bestmaintenance policy. When we do some research on the repairable problems,we often need some statistical methods, such as parameter estimation,interval estimation and hypothesis testing, to consider the parameter andinterval estimation for the components’ lifetime distribution and thereliability indexes of the systems. In practice, the results can be used to makesome decision on the choice of the components’ lifetime distribution.Furthermore, the results can also be used as a standard to evaluate thereliability and the stability of the systems. So, it is of great significance andapplication value to study the statistical problems of the repairable systems.In this article, we study several statistical inference problems of therepairable system and then consider a δ-shock maintenance model for arepairable deteriorating system.Firstly, we research on the estimation problems for the three important parameters of an alpha-series process, while the distributions of the firstarriving time are lognormal distribution and truncated normal distribution.We obtain the three parameters’ different kinds of estimators and consider theconsistency and asymptotic normality properties of some of the estimators.And then we compare the performance of these variety estimators based onthe simulation results.Secondly, we study the statistical inference for the lifetime of warmstandby system. As the switch is not entirely, we get the moment estimation,the maximum likelihood estimation, the inverse moment estimation and theconfidence lower limit for the lifetime of the system, while the operating timefollows the gamma distribution and the repairing time follows the gammadistribution or the Weibull distribution.Thirdly, we study the confidence estimation of the steady stateavailability for a repairable system while the operating time follows thelognormal distribution and the repairing time follows the gamma distribution.We get the confidence estimations by the methods of pivot and the one-sidedor two-sided likelihood test.What’s more, we study the Bayes estimation of the parameters of thelognormal distribution and the Inverse Gaussian distribution. We get the Bayes estimation of the parameters and the parameter’s loss function and riskfunction, and then we discuss the conservatism property of the loss functionand the risk function.Finally, we research on the maintenance cost problem of a δ-shockmaintenance model for a repairable deteriorating system, in which we use thegeometric process to represent the deteriorating property of the system. Whilethe internal arriving time of the shock obeys to three different kinds ofdistributions, we obtain the long-run average cost per unit time of the system.