u_{\alpha\alpha}=\dfrac{\partial^2 u}{\partial\alpha^2},\quad u_{\beta\beta}=\dfrac{\partial^2 u}{\partial\beta^2},\quad u_{\alpha\beta}=\dfrac{\partial^2 u}{\partial\alpha\partial\beta},
 \end{aligned}$$ $$\tau_{\alpha\alpha}=\dfrac{1}{l_{20}},\,\tau_{\alpha\beta}=\dfrac{1}{l_{11}}=\tau_{\beta\alpha},\, \tau_{\beta\beta}=\dfrac{1}{l_{02}}$$

since, $u(\alpha,\beta$ is the function of $\alpha,\beta$ both. Therefore,

-   If $u(\alpha,\beta)=\alpha$ in (5.12) then; $$\begin{aligned}
     u_\alpha&=1,\quad u_\beta=0, \quad u_{\alpha\alpha}=u_{\beta\beta}=0,\quad u_{\alpha\beta}=u_{\beta\alpha}=0
     \end{aligned}$$

-   If $u(\alpha,\beta)=\beta$ in (5.12) then; $$\begin{aligned}
     u_\beta&=1,\quad u_\alpha=0, \quad u_{\alpha\alpha}=u_{\beta\beta}=0,\quad u_{\alpha\beta}=u_{\beta\alpha}=0\\
     \end{aligned}$$

and the rest derivatives based on likelihood function are as obtained as; $$l_{30}=\dfrac{3n}{\alpha^3},~~l_{11}=-2\sum_{i=1}^{n}x_i^{2\beta}\ln x_i, ~~
l_{03}=\dfrac{2n}{\beta^3}-8\alpha\sum_{i=1}^{n}x_i^{2\beta}(\ln x_i)^{3}$$ $$l_{12}=-4\sum_{i=1}^{n}x_i^{2\beta}(\ln x_i)^{2}=l_{21}$$ Using these derivatives the Bayes estimates of $(\alpha,\beta)$ are obtained by following expressions $$\begin{split}
\hat{\alpha}_{bl}=&\hat{\alpha}_{ml}+\dfrac{1}{2}[(2u_\alpha\rho_\alpha)\tau_{\alpha\alpha}+(2u_\alpha\rho_\beta)\tau_{\alpha\beta}]+\dfrac{1}{2}[(u_\alpha\tau_{\alpha\alpha})(l_{30}\tau_{\alpha\alpha}+2l_{21}\tau_{\alpha\beta}+l_{12}\tau_{\beta\beta})\\&+(u_\alpha\tau_{\beta\alpha})(l_{21}\tau_{\alpha\alpha}+2l_{12}\tau_{\beta\alpha}+l_{03}\tau_{\beta\beta})]
\end{split}$$ $$\begin{split}
\hat{\beta}_{bl}&=\hat{\beta}_{ml}+\dfrac{1}{2}[(2u_\beta\rho_\alpha)\tau_{\beta\alpha}+(2u_\beta\rho_\beta)\tau_{\beta\beta}]+\dfrac{1}{2}[(u_\beta\tau_{\alpha\beta})(l_{30}\tau_{\alpha\alpha}+2l_{21}\tau_{\alpha\beta}+l_{12}\tau_{\beta\beta})\\&+(u_\beta\tau_{\beta\beta})(l_{21}\tau_{\alpha\alpha}+2l_{12}\tau_{\beta\alpha}+l_{03}\tau_{\beta\beta})]
\end{split}$$

Simulation Study
================

In this section, Monte Carlo simulation study has been performed to assess the performance of the obtained estimators in terms of their mean square error (MSEs). The maximum likelihood estimates of the parameters are evaluated by using $nlm () $ function, and MLEs of reliability characteristics are obtained by using invariance properties. The Bayes estimates of the parameter are evaluated by Lindley’s approximation technique. The hyper-parameters values are chosen in such a way that the prior mean is equal to the true value, and prior variance is taken as very small, say 0.5. All the computations are done by $R 3.4.1$ software. At first, we generated 5000 random samples from PMaD $(\alpha,\beta)$ using Newton-Raphson algorithm 