## 4 Bayesian Estimation
### TK Method  
Firstly, the posterior expectation for \(y(\alpha,\lambda)\) is provided by the distribution \(\pi(\alpha,\lambda|\hat{x})\) as follows:  
\[E(y(\alpha,\lambda))=\frac{\int_{0}^{\infty}\int_{0}^{\infty}y( \alpha,\lambda)e^{l(\alpha,\lambda|\hat{x})+\ln\pi(\alpha,\lambda)}\pi(\alpha, \lambda)\mathrm{d}\lambda\mathrm{d}\alpha}{\int_{0}^{\infty}\int_{0}^{\infty} e^{l(\alpha,\lambda|\hat{x})+\ln\pi(\alpha,\lambda)}\pi(\alpha,\lambda)\mathrm{d} \lambda\mathrm{d}\alpha}, \tag{31}\]  
where \(l(\alpha,\lambda|\hat{x})\) represents the log-likelihood function and \(\pi(\alpha,\lambda)\) indicates the prior density. For estimating the posterior expectation, we utilize the TK method to address the challenges associated with integrating ratio problems. To derive the explicit function for \(E(y(\alpha,\lambda))\), we will examine the following functions:  
\[o(\alpha,\lambda)=\frac{\ln\pi(\alpha,\lambda)+l(\alpha,\lambda|x)}{n} \tag{32}\]  
and  
\[o^{*}(\alpha,\lambda)=\frac{\ln y(\alpha,\lambda)}{n}+o(\alpha,\lambda). \tag{33}\]  
We individually maximize the two functions, resulting in the estimates \(\left(\widehat{\alpha_{1}},\widehat{\lambda_{1}}\right)\) and \(\left(\widehat{\alpha_{u}},\widehat{\lambda_{u}}\right)\). Subsequently, we approximate \(E(y(\alpha,\lambda))\) as follows:  
\[E(y(\alpha,\lambda))=\sqrt{\frac{|\sum_{y}^{*}|}{|\sum_{y}^{*}|}}e^{\left[no^{ *}_{y}(\widehat{\alpha_{u}}\widehat{\lambda_{u}})-o(\widehat{\alpha_{1}}, \widehat{\lambda_{1}})\right]}. \tag{34}\]In this context, \(|\sum_{w}^{*}|\) and \(|\sum|\) denote the determinants of the negative inverse Hessian matrices associated with \(o(\alpha,\lambda)\) and \(o_{w}^{*}(\alpha,\lambda)\), respectively. The specific action to be taken is as follows:  
\[o(\alpha,\lambda) =\frac{1}{n}[m\ln\alpha+m\ln\lambda-(1+\alpha)\sum_{i=1}^{m}\ln( \lambda x_{i}+1) \tag{35}\] \[-\alpha\sum_{i=1}^{m}R_{i}\ln(\lambda x_{i}+1)+(a-1)\ln\alpha+(p- 1)\ln\lambda-\alpha b-q\lambda], \tag{36}\]  
Now, we obtain the \(|\sum|\) as follows:  
\[|\sum|=\bigg{[}\frac{\partial^{2}o}{\partial\alpha^{2}}\frac{\partial^{2}o}{ \partial\lambda^{2}}-\frac{\partial^{2}o}{\partial\alpha\partial\lambda} \frac{\partial^{2}o}{\partial\alpha\partial\lambda}\bigg{]}_{\alpha=\widehat{ \alpha}_{1},\lambda=\widehat{\lambda}_{1}}^{-1}, \tag{37}\]  
where \(\frac{\partial^{2}o}{\partial\alpha^{2}}\), \(\frac{\partial^{2}o}{\partial\alpha\partial\lambda}\), \(\frac{\partial^{2}o}{\partial\lambda^{2}}\) are provided in Appendix A. In line with the steps outlined above, the elements of \(\sum^{*}\) are provided in Appendix A. As a result, we can derive the \(|\sum^{*}|\) as follows:  
\[|\sum^{*}|=\bigg{[}\frac{\partial^{2}o^{*}}{\partial\lambda^{2}}\frac{\partial ^{2}o^{*}}{\partial\alpha^{2}}-\frac{\partial^{2}o^{*}}{\partial\lambda\partial \alpha}\frac{\partial^{2}o^{*}}{\partial\alpha\partial\lambda}\bigg{]}_{\alpha= \widehat{\alpha}_{u},\lambda=\widehat{\lambda}_{u}}^{-1}. \tag{38}\]  
Based on the calculations outlined above, we will now consider \(y(\alpha,\lambda)\) in relation to both \(\alpha\) and \(\lambda\). The Bayes estimations are subsequently presented according to the SEL function. \[\widehat{\alpha}_{SEL}=\sqrt{\frac{|\sum_{u}^{*}|}{|\sum|}}e^{n[o_{u}^{*}( \widehat{\alpha_{u}})-o(\widehat{\alpha_{1}}\widehat{\lambda_{1}})]}, \tag{39}\]  
\[\widehat{\lambda}_{SEL}=\sqrt{\frac{|\sum_{u}^{*}|}{|\sum|}}e^{n[o_{\lambda}^ {*}(\widehat{\lambda_{u}})-o(\widehat{\alpha_{1}}\widehat{\lambda_{1}})]}. \tag{40}\]  
Further investigation into Bayesian estimation allows us to derive estimates using the BSEL function. The specific BSEL function is articulated as follows: \(\sigma_{BSEL}=\tau\sigma_{0}+(1-\tau)\sigma_{SEL}\), where \(0\leq\tau\leq 1\).