we took most frequently used square error loss function (SELF) to obtain the estimates of the parameters. It is defined as; $$L(\phi,\hat{\phi})\propto \left( \phi-\hat{\phi}\right)^{2}$$ where, $\hat{\phi}$ is estimate of $\phi$. Bayes estimates under SELF is the posterior mean and evaluated by $$\hat{\phi}_{SELF}=\left[ E(\phi|x)\right]$$ provided the expectation exist and finite. Thus, the Bayes estimator based on equation no. (5.6) under SELF are given by $$\hat{\alpha}_{bs}=E_{\alpha,\beta|x}(\alpha|\beta,x)=\eta \int_{\alpha}\int_{\beta}\alpha^{\frac{3}{2}+a}\beta^{n+c-1} e^{-\alpha \left(b+\sum_{i=1}^{n} x_i^{2\beta}\right) }e^{-d\beta}\left( \prod_{i=1}^{n}x_i^{3\beta-1}\right)  ~ d\alpha~ d\beta$$ and $$\hat{\beta}_{bs}=E_{\alpha,\beta|x}(\beta|\alpha,x)=\eta \int_{\alpha}\int_{\beta}\alpha^{\frac{1}{2}+a}\beta^{n+c} e^{-\alpha \left(b+\sum_{i=1}^{n} x_i^{2\beta}\right) }e^{-d\beta}\left( \prod_{i=1}^{n}x_i^{3\beta-1}\right)  ~ d\alpha~ d\beta$$ where, $\eta=\int_{\alpha}\int_{\beta}\alpha^{\frac{3n}{2}+a-1}\beta^{n+c-1} e^{-\alpha \left(b+\sum_{i=1}^{n} x_i^{2\beta}\right) }e^{-d\beta}\left( \prod_{i=1}^{n}x_i^{3\beta-1}\right) ~d\alpha~ d\beta$\
From equation number (5.9), (5.10) it is easy to observed that the posterior expectations are appearing in the form of the ratio of two integrals. Thus, the analytical solution of these expetations are not presumable. Therefore, any numerical approximation techniques may be implemented to secure the solutions. Here, we used one of the most popular and quite effective approximation technique suggested by Lindley (1980). The detailed description can be seen in below; $$\begin{aligned}
\label{eq20}
 (\hat{\alpha},\hat{\beta})_{Bayes}&=&\dfrac{\int_{\alpha}\int_{\beta}u(\alpha,\beta)e^{\rho(\alpha,\beta)+l}\quad d\alpha d\beta }{\int_{\alpha}\int_{\beta}e^{\rho(\alpha,\beta)+l}\quad d\alpha d\beta}\\&=&(\hat{\alpha},\hat{\beta})_{ml}+\dfrac{1}{2}[(u_{\alpha\alpha}+ 2 u_\alpha\rho_\alpha)\tau_{\alpha\alpha}+(u_{\alpha\beta}+2u_\alpha\rho_\beta)\tau_{\alpha\beta}+(u_{\beta\alpha}+ 2u_\beta\rho_\alpha)\tau_{\beta\alpha}\nonumber\\
 &+&(u_{\beta\beta}+2u_\beta\rho_\beta)\tau_{\beta\beta}]+\dfrac{\alpha}{\beta}[(u_\alpha\tau_{\alpha\alpha}+u_\beta\tau_{\alpha\beta})(l_{111}\tau_{\alpha\alpha}+2l_{21}\tau_{\alpha\beta}+l_{12}\tau_{\beta\beta})\nonumber\\
 &+&(u_\alpha\tau_{\beta\alpha}+u_\beta\tau_{\beta\beta})(l_{21}\tau_{\alpha\alpha}+2 l_{12}\tau_{\beta\alpha}+l_{222}\tau_{\beta\beta})]
 \end{aligned}$$ where, $u(\alpha,\beta)=(\alpha,\beta)$, $\rho(\alpha,\beta)=\ln g(\alpha,\beta)$ and $l=\ln L(\alpha,\beta|\underbar x)$, $$\begin{aligned}
    l_{ab}=\dfrac{\partial^{3}l}{\partial\alpha^{a}\partial\beta^{b}},\quad a,b=0,1,2,3\quad a+b=3,\quad
    \rho_\alpha=\dfrac{\partial \rho}{\partial\alpha},\quad\rho_\beta=\dfrac{\partial \rho}{\partial\beta}\\ u_\alpha=\dfrac{\partial u}{\partial\alpha},\quad u_\beta=\dfrac{\partial u}{\partial \beta},\quad 