\\
l_{\beta\alpha} & l_{\beta\beta}
\end{pmatrix}_{(\hat{\alpha},\hat{\beta})}  \eqno{(5.2.1)}$$ where, $$l_{\alpha\alpha}=-\dfrac{3n}{2\alpha^2},~~l_{\alpha\beta}=-2\sum_{i=1}^{n}x_i^{2\beta}\ln x_i, ~~
l_{\beta\beta}=-\dfrac{n}{\beta^2}-4\alpha\sum_{i=1}^{n}x_i^{2\beta}(\ln x_i)^{2}$$

The above matrix can be inverted and diagonal elements of $I^{-1}(\hat{\alpha},\hat{\beta})$ provide asymptotic variance of $\alpha$ and $\beta$ respectively. Now, two sided $100(1-\Psi)\%$ asymptotic confidence interval for $\alpha$, $\beta$ has been obtained as $$[\alpha_l,\alpha_u] \in [\hat{\alpha}\mp Z_{1-\frac{\Psi}{2}}\sqrt{var(\hat{\alpha})}]$$ $$[\beta_l,\beta_u] \in [\hat{\beta}\mp Z_{1-\frac{\Psi}{2}}\sqrt{var(\hat{\beta})}]$$ respectively.

Bayes Estimation
----------------

In this subsection, the Bayes estimation procedure for the PMaD has been developed. In this estimation technique, as we all know that the unknown parameter treated as the random variable and this randomness of the parameter quantify in the form of prior distribution. Here, we took two independent gamma prior for both shape and scale parameter. The considered prior is very flexible due to its flexibility of assuming different shape. Thus, the joint prior $g(\alpha,\beta)$ is given by;

$$g(\alpha,\beta)\propto \alpha^{a-1}\beta^{c-1}e^{-b\alpha-d\beta}~~;~~~\alpha, \beta>0$$

where, a, b, c & d are the hyperparmaters of the considered priors. Using (5.1) and (5.5), the joint posterior density function $\pi(\alpha,\beta|x)$ is derived as $$\begin{split}
\pi(\alpha,\beta|x)&= \dfrac{L(x|\alpha,\beta)g(\alpha,\beta)}{\int_{\alpha}\int_{\beta} L(x|\alpha,\beta)g(\alpha,\beta) d\alpha\,d\beta}\\&= \dfrac{\alpha^{\frac{3n}{2}+a-1}\beta^{n+c-1} e^{-\alpha \left(b+\sum_{i=1}^{n} x_i^{2\beta}\right) }e^{-d\beta}\left( \prod_{i=1}^{n}x_i^{3\beta-1}\right) }{\int_{\alpha}\int_{\beta}\alpha^{\frac{3n}{2}+a-1}\beta^{n+c-1} e^{-\alpha \left(b+\sum_{i=1}^{n} x_i^{2\beta}\right) }e^{-d\beta}\left( \prod_{i=1}^{n}x_i^{3\beta-1}\right) \quad d\alpha\quad d\beta}
\end{split}$$ In the Bayesian analysis, the specification of proper loss function plays an important role. Here, 