Statistical inference on multi-component stress-strength parameter in three parameter Weibull distribution

The statistical inference of the multi-component stress-strength parameter, $R_{s,k}$, is considered in the three-parameter Weibull distribution. The problem is studied in two cases. In the first case, assuming that the stress and strength variables have common shape and location parameters and non-common scale parameters and all these parameters are unknown, the maximum likelihood estimation and the Bayesian estimation of the parameter $‎R_{s,k}$ are investigated. In this case, as the Bayesian estimation does not have a closed form, it is approximated by two methods, Lindley and $mbox{MCMC}$. Also, asymptotic confidence intervals have been obtained. In the second case, assuming that the stress and strength variables have known common shape and location parameters and non-common and unknown scale parameters, the maximum likelihood estimation, the uniformly minimum variance unbiased estimators, the exact Bayesian estimation of the parameter $‎R_{s,k}$ and the asymptotic confidence interval is calculated. Finally, using Monte Carlo simulation, the performance of different estimators has been compared.