## 3 Interval Estimation
### Bootstrap Confidence Interval
To compute the Bayesian estimates, three priors are considered. The non-informative prior is denoted as prior 1 and the gamma priors are denoted as prior 2 for \(\eta_{1}\) and prior 3 for \(\eta_{2}\). The relevant parameters for the priors are as follows:

Prior 1: \(a_{i}=b_{i}=0,\ i=1,\ 2,\ 3\). Prior 2: \(a_{1}=1,\ b_{1}=1,\ a_{2}=1,\ b_{2}=1,\ a_{3}=2,\ b_{3}=1\). Prior 3: \(a_{1}=3,\ b_{1}=2,\ a_{2}=2,\ b_{2}=1,\ a_{3}=2,\ b_{3}=1\). For the GELF, three loss functions are considered:

F1: squared error loss function \((q=-1)\)

F2: entropy loss function \((q=1)\)

F3: precautionary loss function \((q=-2)\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**(\(\boldsymbol{k}\), \(\boldsymbol{K}\))** & **C.S** & **(\(\boldsymbol{u}\), \(\boldsymbol{N}\))** & **C.S** \\ \hline \((5,15)\) & \(S_{1}\) & \((0\times 4,10)\) & \((10,15)\) & \(T_{1}\) & \((0\times 9,5)\) \\ \(S_{2}\) & \((10,0\times 4)\) & & \(T_{2}\) & \((5,0\times 9)\) \\ \(S_{3}\) & \((2\times 5)\) & & \(T_{3}\) & \(((0,1)\times 5)\) \\ \((4,10)\) & \(S_{4}\) & \((0,0,0,6)\) & \((15,30)\) & \(T_{4}\) & \((0\times 14,15)\) \\ \(S_{5}\) & \((6,0,0,0)\) & & \(T_{5}\) & \((15,0\times 14)\) \\ \(S_{6}\) & \((2,2,1,1)\) & & \(T_{6}\) & \((1\times 15)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Different censoring schemes. For two different sets of parameters and multiple combinations of censoring schemes, we obtain the MLEs and Bayes estimates. Then, we compute their mean absolute deviations (MAD) and mean squared errors (MSE), as shown in Table 2 for \(\eta_{1}\) and Table 3 for \(\eta_{2}\). The ACIs, BootCIs, BCIs, and HPDCIs for \(R_{s,k}\) at a 95% confidence level are also derived under the same combinations of censoring schemes for two different sets of parameters. When computing the BootCIs, we take \(B=1000\) re-samples. The average lengths (AL) and the coverage probabilities (CP) for different types of intervals are shown in Table 4 for \(\eta_{1}\), and Table 5 for \(\eta_{2}\). To ensure the feasibility of the MCMC method, the posterior density plots for \(\lambda\) are provided, as shown in Figures 3 and 4. It can be observed that these posterior density plots are very close to the density plots of the Gaussian distributions. Therefore, when implementing the M-H algorithm to obtain the MCMC samples, it is possible to use the Gaussian distribution as the proposed density. We perform \(T=\)10,000 iterations for the MCMC method. To ensure convergence, trace plots for the three parameters are provided, as illustrated in Figures 5 and 6. As can be seen in these two figures, the MCMC chains rapidly converge to their stationary distributions. Thus, we can consider \(T_{0}=500\) as a burn-in period. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Bayes**} \\ \hline  & \multicolumn{6}{c}{**MLE**} & \multicolumn{6}{c}{**Prior 1**} & \multicolumn{2}{c}{**Prior 3**} \\ \hline \(\eta_{i}\) & **C.S** & **MAD** & **MSE** & **GELF** & **MAD** & **MSE** & **MAD** & **MSE** \\ \hline \(\eta_{2}\) & (\(S_{1}\), \(T_{1}\)) & 0.0888 & 0.0121 & F1 & 0.0852 & 0.0112 & 0.0719 & 0.0080 \\  & & & & F2 & 0.0921





Figure 3: Posterior density plots of \(\lambda\) for \(\eta_{1}\) (**a**) with C.S (\(S_{3}\), \(T_{6}\)), (**b**) with C.S (\(S_{2}\), \(T_{5}\)).