## 3 Interval Estimation
### Bootstrap Confidence Interval
Nonetheless, the Metropolis-Hastings (M-H) algorithm is utilized to obtain the values of \(\lambda\). The procedure of the Gibbs sampling is provided in Algorithm 2. Then, according to the sampling results, the Bayes estimate with the GELF is

\[\hat{R}_{s,k}^{\text{MC}}=[\frac{1}{T-T_{0}}\sum_{t=T_{0}+1}^{T}\left(R_{s,k}^{ (t)}\right)^{-q}]^{-1/q}, \tag{23}\]

where \(T_{0}\) is the burm-in period. ```
Step 1 Set the initial values \(\theta_{1}{}^{(0)},\theta_{2}{}^{(0)}\), and \(\lambda^{(0)}\). Step 2 Set \(t=1\). Step 3 Using the M-H algorithm, \(\lambda^{(t)}\) can be generated from \(\pi_{1}(\lambda|\theta_{1}{}^{(t-1)},\theta_{2}{}^{(t-1)},data)\). Here, the \(N(\lambda^{(t-1)},1)\) is used as the proposal distribution. Step 4 Obtain \(\theta_{1}{}^{(t)}\) from \(gamma(nk+a_{1},b_{1}+\sum_{t=1}^{n}\sum_{j=1}^{k}\left(e^{\
u_{j}^{(t-1)}}-1 \right)(1+S_{j}))\). Step 5 Obtain \(\theta_{2}{}^{(t)}\) from \(gamma(n+a_{2},b_{2}+\sum_{i=1}^{n}\left(e^{\
u_{i}^{(t-1)}}-1\right)(1+T_{i}))\). Step 6 Compute \(R_{s,k}^{(t)}\) using \(\theta_{1}{}^{(t)}\) and \(\theta_{2}{}^{(t)}\). Step 7 Set \(t=t+1\). Step 8 Repeat Steps 3-7 \(T\) times. ```

**Algorithm 2** The algorithm of the MCMC method. ### Bayesian Credible Intervals and Highest Posterior Density Credible Intervals

In this subsection, the BCI and HPDCI are constructed in the following way:

To begin with, the generated posterior samples \(R_{s,k}^{(t)}\), \(t=T_{0}+1\), \(T_{0}+2\),..., \(T\) are sorted to derive the values \(R_{(1)}\leq R_{(2)}\leq...\leq R_{(T-T_{o})}\). Then, the BCI with a \(100(1-\alpha)\%\) confidence level for \(R_{s,k}\) is obtained as

\[(R_{([(T-T_{0})(\alpha/2)])^{\prime}},\ R_{([(T-T_{0})(1-\alpha/2)])}).\]

Among all the possible intervals with a \(100(1-\alpha)\%\) confidence level, the HPDCI has the shortest width. Thus, the HPDCI for \(R_{s,k}\) is given by

\[(R_{(n)},\ R_{(n+[(T-T_{0})(1-\alpha)])}),\]

where \(n\) is the integer that makes

\[R_{(n+[(T-T_{0})(1-\alpha)])}-R_{(n)}=\min_{1\leq n\leq T-T_{0}}R_{(n+[(T-T_{0 })(1-\alpha)])}-R_{(n)},\]

where \([x]\) is the least integer function. ## 5 Numerical Explorations

### Simulation Studies

To analyze the effects of various estimates, Monte Carlo simulations are performed. First, the samples with the censoring scheme \(\{N,\ n,\ S_{1},\ S_{2},\...,\ S_{n}\}\) are generated according to Algorithm 3, as discussed by [PERSON] and [PERSON] [22]. For the different point estimates, including MLEs and Bayes estimates, we compute their corresponding mean absolute deviations and mean squared errors. For different interval estimates, including 95% ACIs, BootCIs, BCIs, and HPDCIs, we compute their corresponding average lengths and coverage probabilities. The various progressive Type-II censoring schemes (C.S) selected for the simulations are shown in Table 1, where \(S_{1}\), \(S_{2}\),..., \(S_{6}\) are the censoring schemes used for the strength variables, and \(T_{1}\), \(T_{2}\),..., \(T_{6}\) are the censoring schemes used for the stress variables. We perform lifetime tests on \(N\) systems with \(K\) components and observe the data obtained for \(n\) systems with \(k\) components using a certain censoring scheme. For simplicity, a short notation such as (\(1\times 3\)) represents (\(1\), \(1\), \(1\)) and (\((0,1)\times 3\)) represents (\(0\), \(1\), \(0\), \(1\), \(0\), \(1\)). Take \(s=3\) in schemes \(S_{1}\), \(S_{2}\), and \(S_{3}\) and \(s=2\) in schemes \(S_{4}\), \(S_{5}\), and \(S_{6}\). When at least two or three components survive, we derive the estimations of the system reliability. The simulation results are based on 3000 replications and all simulation calculations are performed using R-4.3.0 software. Here, two different sets of distribution parameters are used:

\[\eta_{1}=(\theta_{1},\ \theta_{2},\ \lambda)=(1,\ 1,\ 2)\,\ \eta_{2}=(\theta_{1},\ \theta_{2},\ \lambda)=(1.5,\ 2,\ 2).\]

For \(\eta_{1}\), \(R_{3,5}\) is \(0.5\) and \(R_{2,4}\) is \(0.6\), whereas for \(\eta_{2}\), \(R_{3,5}\) is \(0.5901\) and \(R_{2,4}\) is \(0.6885\).