statistical inference of component lifetimes in a coherent system under proportional hazard rate model with known signature

In this paper‎, ‎we discuss the statistical inference of the lifetime distribution of components in a $n$-component coherent system when the system structure is known and the component lifetime follows the proportional hazard rate model‎. ‎different estimation methods‎, ‎the maximum likelihood estimator‎, ‎approximation of the maximum likelihood estimator‎, ‎and bayes estimator for the component lifetime parameter are discussed‎. ‎because the integrals of the bayes estimates do not possess closed forms‎, ‎the metropolis-hastings method and lindley's approximate method are applied to approximate these integrals‎. ‎confidence intervals based on the asymptotic distribution of the mle‎, ‎likelihood ratio test‎, ‎pivotal method‎, ‎and highest posterior density credible are computed‎. ‎two numerical examples are used to illustrate the methodologies developed in this paper and a monte carlo simulation study is used to compare the performance of these estimation methods and recommendations are made based on these results.