Estimation of reliability in a multicomponent stress–strength model based on a bivariate Kumaraswamy distribution

In this paper, we consider a system which has k statistically independent and identically distributed strength components and each component is constructed by a pair of statistically dependent elements. These elements (X1,Y1),(X2,Y2),…,(Xk,Yk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (X_{1},Y_{1}),(X_{2},Y_{2}),\ldots ,(X_{k},Y_{k})$$\end{document} follow a bivariate Kumaraswamy distribution and each element is exposed to a common random stress T which follows a Kumaraswamy distribution. The system is regarded as operating only if at least s out of k(1≤s≤k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1\le s\le k)$$\end{document} strength variables exceed the random stress. The multicomponent reliability of the system is given by Rs,k=P(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{s,k}=P($$\end{document}at least s of the (Z1,…,Zk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Z_{1},\ldots ,Z_{k})$$\end{document} exceed T) where Zi=min(Xi,Yi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{i}=\min (X_{i},Y_{i})$$\end{document}, i=1,…,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,k$$\end{document}. We estimate Rs,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_{s,k}$$\end{document} by using frequentist and Bayesian approaches. The Bayes estimates of Rs,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{s,k}$$\end{document} have been developed by using Lindley’s approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms. The uniformly minimum variance unbiased and exact Bayes estimates of Rs,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{s,k}$$\end{document} are obtained analytically when the common second shape parameter is known. The asymptotic confidence interval and the highest probability density credible interval are constructed for Rs,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{s,k}$$\end{document}. The reliability estimators are compared by using the estimated risks through Monte Carlo simulations. Real data are analysed for an illustration of the findings.