## 4 Bayesian Method  
### Prior Information  
For this part, we assume independent gamma distributions as the prior distributions for the three random variables (\(\alpha\), \(\lambda\), and \(\beta-1\)). That is, \(\alpha\sim\text{Gamma}(a_{1},b_{1})\), \(\lambda\sim\text{Gamma}(a_{2},b_{2})\) and \(\beta-1\sim\text{Gamma}(a_{3},b_{3})\). After that, the joint prior density \(\pi(\alpha,\lambda,\beta)\) is formulated as shown below:  
\[\pi(\alpha,\lambda,\beta)\propto\alpha^{a_{1}-1}\lambda^{a_{2}-1}(\beta-1)^{a_ {3}-1}e^{-(b_{1}\alpha+b_{2}\lambda+b_{3}(\beta-1))}, \tag{10}\]  
where \(a_{1},b_{1},a_{2},b_{2}\) and \(a_{3},b_{3}\) are hyper-parameters. They are non-negative and pre-determined. Independent gamma distributions has been chosen as the priors because of the adaptable nature of the gamma distribution.  
Then the joint posterior density function is derived with regard for the formulated likelihood and prior density functions:  
\[\begin{split}\pi(\alpha,\lambda,\beta\mid\mathbf{x})=& C^{-1}\alpha^{m_{1}+m_{2}-r_{1}-r_{2}+a_{1}-1}\lambda^{m_{1}+m_{2}-r_{1}-r_{2}+a_{ 2}-1}\beta^{m_{2}-r_{2}}(\beta-1)^{a_{3}-1}e^{-(b_{1}\alpha+b_{2}\lambda+b_{3}( \beta-1))}\\ &\times\left[1-\left(1-e^{-\lambda/x_{1}r_{1}+1}\right)^{\alpha} \right]^{r_{1}}\left[1-\left(1-e^{-\lambda/x_{2}r_{2}+1}\right)^{\beta\alpha} \right]^{r_{2}}\\ &\times\prod_{i=r_{1}+1}^{m_{1}}\frac{1}{x_{1i}^{2}}e^{-\lambda/x _{1i}}\Big{(}1-e^{-\lambda/x_{1i}}\Big{)}^{\alpha R_{1i}+\alpha-1}\\ &\times\prod_{i=r_{2}+1}^{m_{2}}\frac{1}{x_{2i}^{2}}e^{-\lambda/x _{2i}}\Big{(}1-e^{-\lambda/x_{2i}}\Big{)}^{\beta\alpha R_{2}+\beta\alpha-1}, \end{split} \tag{11}\]  
where \(\mathbf{x}\) denotes the data, and\[\begin{split} C=\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}& \alpha^{m_{1}+m_{2}-r_{1}-r_{2}+a_{1}-1}\lambda^{m_{1}+m_{2}-r_{1}-r_{2}+a_{2 }-1}\beta^{m_{2}-r_{2}}(\beta-1)^{a_{3}-1}e^{-(b_{1}a+b_{2}\lambda+b_{3}(\beta -1))}\\ &\times\Big{[}1-\Big{(}1-e^{-\lambda/x_{1}\gamma_{1}+1}\Big{)}^{ \alpha}\Big{]}^{r_{1}}\Big{[}1-\Big{(}1-e^{-\lambda/x_{2}\gamma_{2}+1}\Big{)} ^{\beta\alpha}\Big{]}^{r_{2}}\\ &\times\prod_{i=r_{1}+1}^{m_{1}}\frac{1}{x_{1i}^{2}}e^{-\lambda/ x_{1i}}\Big{(}1-e^{-\lambda/x_{1i}}\Big{)}^{\alpha R_{1i}+\alpha-1}\\ &\times\prod_{i=r_{2}+1}^{m_{2}}\frac{1}{x_{2i}^{2}}e^{-\lambda/ x_{2i}}\Big{(}1-e^{-\lambda/x_{2i}}\Big{)}^{\beta\alpha R_{2i}+\beta\alpha-1}d \alpha d\lambda d\beta.\end{split} \tag{12}\]  
Thereafter, the formula for obtaining the Bayesian estimate of an arbitrary function \(h(\alpha,\lambda,\beta)\) using the squared error loss function (SELF) may be provided by:  
\[\begin{split}\tilde{h}(\alpha,\lambda,\beta)=& \mathrm{E}_{\alpha,\lambda,\beta|x}[h(\alpha,\lambda,\beta)]\\ =&\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty }h(\alpha,\lambda,\beta)\pi(\alpha,\lambda,\beta\mid x)d\alpha d\lambda d\beta,\end{split} \tag{13}\]  
obviously it is impossible to compute (13), it is for this reason that the importance sampling approach is then utilized to establish the Bayesian point estimates. Additionally, we employ the same methodology to create the Bayesian credible intervals (BCIs) and the highest posterior density credible intervals (HPDCIs).