l}{\partial\beta}=0$$ which yield, $$\dfrac{3n}{2\alpha}-\sum_{i=1}^{n}x_i^{2\beta}=0$$ $$\dfrac{n}{\beta}-2\alpha\sum_{i=1}^{n} x_i^{2\beta}\ln x_i+3\sum_{i=1}^{n}\ln x_i =0$$ The MLE’s of the parameters are obtained by solving the above two equations simultaneously. Here, we used non-linear maximization techniques to get the solution.

### Uniqueness of MLEs

The uniqueness of MLEs discussed in previous section can be checked by using following propositions.\
**Proposition 1:** If $\beta$ is fixed, then $\hat{\alpha}$ exist and it is unique.\
**Proof:** Let $L_{\alpha}=\dfrac{3n}{2\alpha}-\sum_{i=1}^{n}x_i^{2\beta}$, since $L_{\alpha}$ is continuous and it has been verified that $\lim\limits_{\alpha\rightarrow 0}L_{\alpha}=\infty$ and $\lim\limits_{\alpha\rightarrow \infty}L_{\alpha}=-\sum_{i=1}^{n}x_i^{2\beta}<0$. This implies that $L_{\alpha}$ will have atleast one root in interval $(0,\infty)$ and hence $L_{\alpha}$ is a decreasing function in $\alpha$. Thus, $L_{\alpha}=0$ has a unique solution in $(0,\infty)$.\
**Proposition 2:** If $\alpha$ is fixed, then $\hat{\beta}$ exist and it is unique.\
**Proof:** Let $L_{\beta}=\dfrac{n}{\beta}-\alpha\sum_{i=1}^{n} x_i^{2\beta}\ln x_i+3\sum_{i=1}^{n}\ln x_i $, since $L_{\beta}$ is continuous and it has been verified that $\lim\limits_{\beta\rightarrow 0}L_{\beta}=\infty$ and $\lim\limits_{\beta\rightarrow \infty}L_{\beta}=-2\sum_{i=1}^{n}\ln {x_i}<0$. This implies in same as above $\hat{\beta}$ exists and it will be unique.

### Fisher Information Matrix

Here, we derive Fisher information matrix for constructing 100$(1-\Psi)\%$ asymptotic confidence interval for the parameters using large sample theory. The Fisher information matrix can be obtained by using equation (5.2) as $$I(\hat{\alpha},\hat{\beta})=-E\begin{pmatrix}
l_{\alpha\alpha} & l_{\alpha\beta}\\
