## 3 Interval Estimation
### Bootstrap Confidence Interval  
The MLEs may not follow the asymptotic normality when the observed sample size is insufficient, which can be a limitation of the ACI. Therefore, the BootCI is considered. Using the method discussed by [PERSON] [20], the percentile BootCI is constructed. The procedures are provided in Algorithm 1. ```
Step 1 Obtain the MLEs \(\hat{\theta}_{1}\), \(\hat{\theta}_{2}\), and \(\hat{\lambda}\) with the given data sets \(\underset{\sim}{x}\) and \(\underset{\sim}{y}\) using the method mentioned in Section 2. Step 2 For a fixed progressive censoring scheme \((S_{1},\ S_{2},\...,\ S_{k})\), generate censored samples \(\ \



## 4 Bayesian Estimation

In this section, Bayesian estimation is discussed, together with the generalized entropy loss function (GELF). When overestimation and underestimation are considered different consequences of errors, the GELF is useful. The GELF is a suitable loss function used for reliability estimation since overestimating is typically considerably more harmful than underestimating. As first discussed by [PERSON] and [PERSON] [21], the GELF is given by

\[L(\alpha,\hat{\alpha})\propto[(\frac{\hat{\alpha}}{\hat{\alpha}})^{q}-q\ln( \frac{\hat{\alpha}}{\hat{\alpha}})-1];q\
eq 0,\]

where \(\hat{\alpha}\) is the decision rule to estimate \(\alpha\). The Bayes estimate with the GELF is derived as

\[\hat{\alpha}=[E(\alpha^{-q}|data)]^{-1/q}. \tag{20}\]

Note that the Bayes estimate in Equation (20) corresponds to the Bayes estimate with a precautionary loss function when \(q=-2\), a squared error loss function when \(q=-1\), and an entropy loss function when \(q=1\). In Bayesian estimation, \(\theta_{1}\), \(\theta_{2}\), and \(\lambda\) can be considered random variables. Assume that they have independent gamma priors and their PDFs are expressed as follows:

\[ \begin{split} g_{1}(\theta_{1})\propto\theta_{1}{}^{a_{1}-1}e^{- b_{1}\theta_{1}},\ \theta_{1}>0,\\ g_{2}(\theta_{2})\propto\theta_{2}{}^{a_{2}-1}e^{-b_{2}\theta_{2} },\ \theta_{2}>0,\\ g_{3}(\lambda)\propto\lambda^{a_{3}-1}e^{-b_{3}\lambda},\ \lambda>0, \end{split} \]

where \(a_{i},\ b_{i}>0,\ i=1,\ 2,\ 3\). Then, the joint posterior density function is

\[\pi(\theta_{1},\theta_{2},\lambda|data)=\frac{L(data|\theta_{1},\theta_{2}, \lambda)g_{1}(\theta_{1})g_{2}(\theta_{2})g_{3}(\lambda)}{\int_{0}^{\infty} \int_{0}^{\infty}\int_{0}^{\infty}L(data|\theta_{1},\theta_{2},\lambda)g_{1}( \theta_{1})g_{2}(\theta_{2})g_{3}(\lambda)d\theta_{1}d\theta_{2}d\lambda}. \tag{21}\]

By substituting Equation (6) into Equation (21), the posterior distribution is

\[ \begin{split}\pi(\theta_{1},\theta_{2},\lambda|data)\propto& \theta_{1}{}^{nk+a_{1}-1}\theta_{2}{}^{n+a_{2}-1}\lambda^{n(k+1)+a_{3}-1}e^{ -(b_{1}\theta_{1}+b_{2}\theta_{2}+b_{3}\lambda)}\\ &\times\prod_{i=1}^{n}\prod_{j=1}^{k}x_{ij}{}^{\lambda-1}\exp\{x_ {ij}{}^{\lambda}+\theta_{1}(1-e^{x_{ij}^{\lambda}})(1+S_{j})\}\\ &\times\prod_{i=1}^{n}y_{i}{}^{\lambda-1}\exp\{y_{i}{}^{\lambda} +\theta_{2}(1-e^{y_{i}^{\lambda}})(1+T_{i})\}.\end{split}  \tag{22}\]

### MCMC Method

The MCMC method is considered to obtain the Bayesian estimation. According to Equation (22), the conditional posterior density of \(\theta_{1}\), \(\theta_{2}\), and \(\lambda\) are derived as

\[ \begin{split}\theta_{1}|\lambda,data\sim gamma(nk+a_{1},b_{1}+ \sum_{i=1}^{n}\sum_{j=1}^{k}\left(e^{x_{ij}^{\lambda}}-1\right)(1+S_{j})),\\ \theta_{2}|\lambda,data\sim gamma(n+a_{2},b_{2}+\sum_{i=1}^{n} \left(e^{y_{i}^{\lambda}}-1\right)(1+T_{i})),\end{split} \]

and

\[ \begin{split}\pi_{1}(\lambda|\theta_{1},\theta_{2},data)\propto& \lambda^{n(k+1)+a_{3}-1}e^{-b_{3}\lambda}\prod_{i=1}^{n}\prod_{j=1}^{k}x_{ij}{}^ {\lambda}\exp\{x_{ij}{}^{\lambda}+\theta_{1}(1-e^{x_{ij}^{\lambda}})(1+S_{j})\} \\ &\times\prod_{i=1}^{n}y_{i}{}^{\lambda}\exp\{y_{i}{}^{\lambda}+ \theta_{2}(1-e^{y_{i}^{\lambda}})(1+T_{i})\}.\end{split} \]Now, using the gamma distributions, the values of \(\theta_{1}\) and \(\theta_{2}\) can be directly generated.