Estimation of the reliability characteristics by using classical and Bayesian methods of estimation for xgamma distribution

In this article, estimation of the reliability characteristics viz., mean time to system failure M(t;α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(t; \alpha )$$\end{document}, reliability function R(t;α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R(t; \alpha )$$\end{document} is considered for xgamma lifetime distribution. First, four different methods of estimation of the reliability characteristics for specified value of time t are addressed from frequentist approaches and compared them in terms of their respective mean squared errors using extensive numerical simulations. Second, we compared three bootstrap confidence intervals (BCIs) including standard bootstrap, percentile bootstrap and bias-corrected percentile bootstrap. Third, Bayesian estimation is considered under three loss functions using gamma prior for the considered model. Fourth, we obtained highest posterior density (HPD) credible intervals of M(t;α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(t; \alpha )$$\end{document} and R(t;α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R(t; \alpha )$$\end{document}. Monte Carlo simulation study has been carried out to compare the performances of the classical BCIs and HPD credible intervals of M(t;α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(t; \alpha )$$\end{document} and R(t;α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R(t; \alpha )$$\end{document} in terms of average widths and coverage probabilities. Finally, a real data set has been analyzed for illustrative purpose.