## 3 Interval Estimation
### Bootstrap Confidence Interval
Thus, the rejection region is \(\{\Lambda>\chi_{\alpha}^{2}(1)\}\) at a given significance level \(\alpha\), where \(\chi_{\alpha}^{2}(1)\) is \(100(1-\alpha)th\) percentile of \(\chi^{2}(1)\). For given data sets \(X\) and \(Y\), we calculate that \(\Lambda=0.2580\) with a \(p\)-value of 0.6115. Therefore, we can accept \(H_{0}\) at the 0.05 significance level, whereupon the MSSR can be estimated using our proposed method. Then, we obtain different estimates of \(R_{s,k}\) using the complete sample, as well as three different censored samples. The samples can be, respectively, generated by the censoring schemes as follows:

Complete Sample: \(S=(0*5),\ T=(0*7)\quad(n=7,\ k=5,\ s=2)\). C.S 1: \(S=(0,\ 0,\ 0,\ 1),\ T=(0,\ 0,\ 0,\ 1,\ 1)\quad(n=5,\ k=4,\ s=2)\). C.S 2: \(S=(1,\ 0,\ 0,\ 0),\ T=(2,\ 0,\ 0,\ 0,\ 0)\quad(n=5,\ k=4,\ s=2)\). C.S 3: \(S=(1,\ 1,\ 0),\ T=(1,\ 1,\ 1,\ 0)\quad(n=4,\ k=3,\ s=1)\). Based on C.S 1, the censored samples are derived as

\[X_{1}=\left( \begin{array}{cccc}0.291172&0.363359&0.371904&0.414087\\ 0.538082&0.561238&0.722613&0.744881\\ 0.126977&0.287785&0.703119&0.729986\\ 0.294834&0.392917&0.461618&0.613911\\ 0.423813&0.715158&0.726164&0.811159\\ \end{array} \right),\quad Y_{1}=\left( \begin{array}{cccc}0.640395\\ 0.650691\\ 0.667157\\ 0.679829\\ 0.767135\\ \end{array} \right)\] \begin{table}
\begin{tabular}{c c c c c c} \hline
**Data Set** & **Distribution** & \(\hat{\theta}\) & \(\hat{\lambda}\) & **K-S Distance** & \(p\)**-Value** \\ \hline \(X\) & [PERSON] & 4.5946 & 3.5431 & 0.1461 & 0.4050 \\  & Weibull & 3.5428 & 0.6249 & 0.1550 & 0.3344 \\  & Burr XII & 3.7702 & 6.5519 & 0.1612 & 0.2908 \\  & Generalized Rayleigh & 2.2869 & 2.1350 & 0.1552 & 0.3329 \\ \hline \(Y\) & [PERSON] & 2.9026 & 3.6574 & 0.2337 & 0.7630 \\  & [PERSON] & 11.0177 & 0.7358 & 0.2297 & 0.7807 \\  & Burr XII & 11.1770 & 31.7219 & 0.2272 & 0.7913 \\  & Generalized Rayleigh & 1481.5031 & 3.9797 & 0.1966 & 0.9048 \\ \hline \end{tabular}
\end{table} 
Table 6: Comparison of different distributions in K–S tests. Based on C.S 2, the censored samples are derived as

\[X_{2}=\left( \begin{array}{cccc}0.291172&0.371904&0.414087&0.463726\\ 0.294834&0.461618&0.613911&0.688100\\ 0.524947&0.605979&0.668612&0.715850\\ 0.423813&0.726164&0.811159&0.829569\\ 0.345075&0.468782&0.742025&0.767070\\ \end{array} \right),\quad Y_{2}=\left( \begin{array}{cccc}0.640395\\ 0.679829\\ 0.709025\\ 0.767135\\ 0.824860\\ \end{array} \right)\]

Based on C.S 3, the censored samples are derived as

\[X_{3}=\left( \begin{array}{cccc}0.291172&0.371904&0.463726\\ 0.126977&0.703119&0.768563\\ 0.524947&0.605979&0.715850\\ 0.345075&0.468782&0.767070\\ \end{array} \right),\quad Y_{3}=\left( \begin{array}{cccc}0.640395\\ 0.667157\\ 0.709025\\ 0.824860\\ \end{array} \right)\]

For the complete, as well as three progressively censored, samples, we derive various point estimates and 95% interval estimates of \(R_{s,k}\), as shown in Tables 7 and 8, respectively. For the Bayesian estimates, non-informative priors are used. For three parameters in the MCMC method, we provide the CUSUM plots in Figure 9 and the trace plots in Figure 10. As can be seen in these figures, the MCMC chains converged quickly to stationary distributions, indicating that the MCMC method performed well. ## 6 Conclusions

We study the estimation of the MSSR based on the Chen distribution using progressively censored data. The reliability inference of multicomponent stress-strength systems

Figure 10: Trace plots for real data. Figure 9: CUSUM plots for \(\theta_{1}\), \(\theta_{2}\), and \(\lambda\) (**a**) with the complete sample, (**b**) with censoring scheme 1, (**c**) with censoring scheme 2, (**d**) with censoring scheme 3.

has attracted significant interest, with numerous scholars contributing to the field. Progressive Type-II censoring has been widely used in lifetime tests for nearly two decades. The Chen distribution is commonly used to model real-life data in the fields of lifetime analysis and reliability theory. We begin by obtaining the MLE.