On the reliability of a multicomponent series system
Consider a series system consisting of several independent components where the components are assumed to have exponential distributions with unknown and possibly unequal failure rates. The objective here is to infer about the reliability of the whole system at a given time t based on independent samples from each component. In this article, we first consider testing a hypothesis on the reliability of the whole system against a suitable alternative and try to follow the likelihood ratio (LR) method. However, closed expressions for the maximum likelihood estimators (MLEs) of the model parameters are not available under the null (alternative) hypothesis, which hinders the development of the traditional LR test method. To avoid this problem, we have found approximate MLEs of the model parameters under the null (and alternative) hypothesis, which are then used to derive an approximate likelihood ratio (ALR) statistic. Using the ALR statistic, we then find a conditional test procedure whose cut-off point can be found numerically subject to the size condition. Finally, the acceptance region of the test procedure is inverted to obtain a confidence interval for the system reliability.
exponential distribution confidence interval gamma distribution hypothesis testing